Solve the quadratic equations. If an equation has no real roots, state this. In cases where the solutions involve radicals, give both the radical form of the answer and a calculator approximation rounded to two decimal places.

$x^{2}+8 x-2=0$

Shamshad W.

Numerade Educator

Solve the quadratic equations. If an equation has no real roots, state this. In cases where the solutions involve radicals, give both the radical form of the answer and a calculator approximation rounded to two decimal places.

$x^{2}-6 x-2=0$

Shamshad W.

Numerade Educator

Solve the quadratic equations. If an equation has no real roots, state this. In cases where the solutions involve radicals, give both the radical form of the answer and a calculator approximation rounded to two decimal places.

$x^{2}+4 x+1=0$

Shamshad W.

Numerade Educator

$x^{2}+12 x+18=0$

Shamshad W.

Numerade Educator

$2 y^{2}-5 y-2=0$

Shamshad W.

Numerade Educator

$3 y^{2}-3 y-4=0$

Shamshad W.

Numerade Educator

$4 y^{2}+8 y+5=0$

Shamshad W.

Numerade Educator

$y^{2}+y+1=0$

Shamshad W.

Numerade Educator

$4 s^{2}-20 s+25=0$

Shamshad W.

Numerade Educator

$16 s^{2}+8 s+1=0$

Shamshad W.

Numerade Educator

$x^{2}=8 x-6$

Shamshad W.

Numerade Educator

$x^{2}=8 x+9$

Shamshad W.

Numerade Educator

$-3 x^{2}+x=-3$

Shamshad W.

Numerade Educator

$(x-5)(x+3)=1$

Shamshad W.

Numerade Educator

$-y^{2}-8 y=1$

Shamshad W.

Numerade Educator

$5 y(y-2)=2$

Shamshad W.

Numerade Educator

$t^{2}=-3 t-4$

Shamshad W.

Numerade Educator

$t^{2}=-3 t+4$

Shamshad W.

Numerade Educator

(a) Use the formula given in Example 1 to estimate the year that the U.S. population reached 100 million.

(b) Use the following U.S. Census Bureau data to say whether your estimate in part (a) is reasonable. In 1910 the population was approximately 92.2 million; in 1920 it was approximately 106.0 million.

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(a) Use the formula given in Example 1 to estimate the year that the U.S. population reached 225 million.

(b) According to the U.S. Census Bureau, the U.S. population in 1980 was 226.5 million. Is your estimate in part (a) reasonable?

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The chart that follows shows the world records for the men's 10,000 meter run in the years 1972 and 1998.

(check your book to see table)

For the years between 1972 and 1998 , the world record can be approximated by the equation

$$y=-0.09781 t^{2}+385.8336 t-378,850.4046$$

$(1972 \leq t \leq 1998)$

where $y$ is the world-record time (in seconds) in the year $t$ (a) Use the given equation, the quadratic formula, and your calculator to estimate the year in which the record might have been 27 minutes $(=1620$ seconds ). (You'll get two solutions for the quadratic; be sure to pick the appropriate one.) Then say by how many years your prediction is off, given the following information: On July $5,1993,$ Richard Chelimo of Kenya ran a record time of 27: 07.91 ; five days after that, Yobes Ondieki, also of Kenya, ran 26: 58.38 .

(b) Estimate the year in which the record might have been $26: 30(=1590$ seconds ). Then say by how many years your prediction is off, given the following information:

On July $4,1997,$ Haile Gebrselassie ran a record time of $27: 07.91 ;$ eighteen days later, Paul Tergat of Kenya $\operatorname{ran} 26: 27.85$

(c) It is interesting to see how accurately (or inaccurately) the quadratic approximating equation predicts record times outside its specified domain. In August 2005 Kenenisa Bekele of Ethiopia ran a record time of 26 minutes 17.54 seconds. (This was still the record in $2008 .$ ) In what year does the equation predict this time would be achieved? How many years off the actual date is the prediction?

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The chart below shows the world records for the women's $10,000$ meter run in the years 1970 and 1993.

(check your book to see table)

For the years between 1970 and $1993,$ the world record can be approximated by the equation

$$\begin{array}{c}y=0.37553 t^{2}-1503.7154 t+1,507,042.699 \\(1970 \leq t \leq 1993)\end{array}$$

where $y$ is the world-record time (in seconds) in the year $t .$ Use the given equation, the quadratic formula, and your calculator to estimate the year in which the record might have been 32 minutes $(=1920$ seconds). (As in Exercise $21,$ you'll get two solutions for the quadratic; pick the appropriate one.) Then say by how many years your prediction is off, given the following information: On September $19,1981,$ Yekena Sipatova of the former Soviet Union ran a record time of $32: 17.2 ;$ in the following year on July $16,$ Mary Decker of the United States ran $31: 35.3 .$ ( Note: Wang Jinxia's 29: 31.78 was still the world record in 2008 .)

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You are given an equation of the form $y=a x^{2}+b x+c$. (a) Use a graphing utility to graph the equation and to estimate the $x$ -intercepts. (Use a zoom-in process to obtain the estimates; keep zooming in until the first three decimal places of the estimate remain the same as you progress to the next step. (b) Determine the exact values of the intercepts by using the quadratic formula. Then use a calculator to evaluate the expressions that you obtain. Round off the results to four decimal places. / Check to see that your results are consistent with the estimates in part (a). $]$

$y=x^{2}-4 x+1$

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You are given an equation of the form $y=a x^{2}+b x+c$. (a) Use a graphing utility to graph the equation and to estimate the $x$ -intercepts. (Use a zoom-in process to obtain the estimates; keep zooming in until the first three decimal places of the estimate remain the same as you progress to the next step. (b) Determine the exact values of the intercepts by using the quadratic formula. Then use a calculator to evaluate the expressions that you obtain. Round off the results to four decimal places. / Check to see that your results are consistent with the estimates in part (a). $]$

$y=x^{2}-10 x+15$

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You are given an equation of the form $y=a x^{2}+b x+c$. (a) Use a graphing utility to graph the equation and to estimate the $x$ -intercepts. (Use a zoom-in process to obtain the estimates; keep zooming in until the first three decimal places of the estimate remain the same as you progress to the next step. (b) Determine the exact values of the intercepts by using the quadratic formula. Then use a calculator to evaluate the expressions that you obtain. Round off the results to four decimal places. / Check to see that your results are consistent with the estimates in part (a). $]$

$y=0.5 x^{2}+8 x-3$

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$y=2 x^{2}+2 x-1$

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$y=2 x^{2}+2 \sqrt{26} x+13$

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$y=3 x^{2}-12 \sqrt{3} x+36$

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Find the sum and the product of the roots of each quadratic equation.

$x^{2}+8 x-20=0$

Shamshad W.

Numerade Educator

Find the sum and the product of the roots of each quadratic equation.

$x^{2}-3 x+12=0$

Shamshad W.

Numerade Educator

Find the sum and the product of the roots of each quadratic equation.

$4 y^{2}-28 y+9=0$

Shamshad W.

Numerade Educator

Find the sum and the product of the roots of each quadratic equation.

$\frac{1}{2} y^{2}=4 y-5$

Shamshad W.

Numerade Educator

Find a quadratic equation with the given roots $r_{1}$ and $r_{2} .$ Write each answer in the form $a x^{2}+b x+c=0,$ where $a, b,$ and $c$ are integers and $a>0$

$r_{1}=3$ and $r_{2}=11$

Shamshad W.

Numerade Educator

Find a quadratic equation with the given roots $r_{1}$ and $r_{2} .$ Write each answer in the form $a x^{2}+b x+c=0,$ where $a, b,$ and $c$ are integers and $a>0$

$r_{1}=-4$ and $r_{2}=-9$

Shamshad W.

Numerade Educator

Find a quadratic equation with the given roots $r_{1}$ and $r_{2} .$ Write each answer in the form $a x^{2}+b x+c=0,$ where $a, b,$ and $c$ are integers and $a>0$

$r_{1}=1-\sqrt{2}$ and $r_{2}=1+\sqrt{2}$

Shamshad W.

Numerade Educator

$r_{1}=2-\sqrt{5}$ and $r_{2}=2+\sqrt{5}$

Shamshad W.

Numerade Educator

$r_{1}=\frac{1}{2}(2+\sqrt{5})$ and $r_{2}=\frac{1}{2}(2-\sqrt{5})$

Shamshad W.

Numerade Educator

$r_{1}=\frac{1}{3}(4-\sqrt{5})$ and $r_{2}=\frac{1}{3}(4+\sqrt{5})$

Shamshad W.

Numerade Educator

Solve the equations. Hint: Look before you leap.

$x^{2}-(\sqrt{2}+\sqrt{5}) x+\sqrt{10}=0$

Shamshad W.

Numerade Educator

Solve the equations. Hint: Look before you leap.

$x^{2}+(\sqrt{2}-1) x=\sqrt{2}$

Shamshad W.

Numerade Educator

A ball is thrown straight upward. Suppose that the height of the ball at time $t$ is given by the formula $h=-16 t^{2}+96 t$ where $h$ is in feet and $t$ is in seconds, with $t=0$ corresponding to the instant that the ball is first tossed.

(a) How long does it take before the ball lands?

(b) At what time is the height 80 ft? Why does this question have two answers?

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During a flu epidemic in a small town, a public health official finds that the total number of people $P$ who have caught the flu after $t$ days is closely approximated by the formula $P=-t^{2}+26 t+106,$ where $1 \leq t \leq 13$

(a) How many have caught the flu after 10 days?

(b) After approximately how many days will 250 people have caught the flu?

Sheryl E.

Numerade Educator

Use the discriminant to determine how many real roots each equation has.

$x^{2}-12 x+16=0$

Shamshad W.

Numerade Educator

Use the discriminant to determine how many real roots each equation has.

$2 x^{2}-6 x+5=0$

Shamshad W.

Numerade Educator

Use the discriminant to determine how many real roots each equation has.

$4 x^{2}-5 x-\frac{1}{2}=0$

Shamshad W.

Numerade Educator

Use the discriminant to determine how many real roots each equation has.

$4 x^{2}-28 x+49=0$

Shamshad W.

Numerade Educator

Use the discriminant to determine how many real roots each equation has.

$x^{2}+\sqrt{3} x+\frac{3}{4}=0$

Shamshad W.

Numerade Educator

Use the discriminant to determine how many real roots each equation has.

$\sqrt{2} x^{2}+\sqrt{3} x+1=0$

Shamshad W.

Numerade Educator

Use the discriminant to determine how many real roots each equation has.

$y^{2}-\sqrt{5} y=-1$

Shamshad W.

Numerade Educator

Use the discriminant to determine how many real roots each equation has.

$\frac{m^{2}}{4}-\frac{4 m}{3}+\frac{16}{9}=0$

Shamshad W.

Numerade Educator

Find the value(s) of $k$ such that the equation has exactly one real root.

$x^{2}+12 x+k=0$

Shamshad W.

Numerade Educator

Find the value(s) of $k$ such that the equation has exactly one real root.

$3 x^{2}+(\sqrt{2 k}) x+6=0$

Shamshad W.

Numerade Educator

Find the value(s) of $k$ such that the equation has exactly one real root.

$x^{2}+k x+5=0$

Shamshad W.

Numerade Educator

Find the value(s) of $k$ such that the equation has exactly one real root.

$k x^{2}+k x+1=0$

Shamshad W.

Numerade Educator

Find the complex roots of the given equation.

$y^{2}+y+1=0$

Shamshad W.

Numerade Educator

Find the complex roots of the given equation.

$4 y^{2}+8 y+5=0$

Shamshad W.

Numerade Educator

Find the complex roots of the given equation.

$t^{2}=-3 t-4$

Shamshad W.

Numerade Educator

Find the complex roots of the given equation.

$x^{2}+100=0$

Shamshad W.

Numerade Educator

Rewrite the equation as $(2 \pi) r^{2}+(2 \pi h) r-20 \pi=0$ and use the quadratic formula with $a=2 \pi, b=2 \pi h,$ and $c=-20 \pi$.

$2 \pi y^{2}+\pi y x=12 ;$ for $y$

Shamshad W.

Numerade Educator

Rewrite the equation as $(2 \pi) r^{2}+(2 \pi h) r-20 \pi=0$ and use the quadratic formula with $a=2 \pi, b=2 \pi h,$ and $c=-20 \pi$.

$-16 t^{2}+v_{0} t=0 ;$ for $t$

Shamshad W.

Numerade Educator

Rewrite the equation as $(2 \pi) r^{2}+(2 \pi h) r-20 \pi=0$ and use the quadratic formula with $a=2 \pi, b=2 \pi h,$ and $c=-20 \pi$.

$-\frac{1}{2} g t^{2}+v_{0} t+h_{0}=0 ;$ for $t$

Shamshad W.

Numerade Educator

(a) $@$ On the same set of axes, graph the equations $y=x^{2}+8 x+16$ and $y=x^{2}-8 x+16$

(b) Use the graphs to estimate the roots of the two equations $x^{2}+8 x+16=0$ and $x^{2}-8 x+16=0 .$ How do the roots appear to be related?

(c) Solve the two equations in part (b) to determine the exact values of the roots. Do your results support the response you gave to the question at the end of part (b)?

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(a) $@$ Figure $1(\mathrm{c})$ in the text shows a graph of the equation $y=x^{2}+3 x-5 .$ Use a graphing utility to reproduce the graph. [Use the same viewing rectangle that is used in Figure $1(\mathrm{c}) .]$

(b) Add the graph of the equation $y=x^{2}-3 x-5$ to the picture that you obtained in part (a). (This new equation is the same as the one in part (a) except that the sign of the coefficient of $x$ has been reversed.) Note that the $x$ -intercepts of the two graphs appear to be negatives of one another.

(c) Use the quadratic formula to determine exact expressions for the roots of the two equations $x^{2}+3 x-5=0$ and $x^{2}-3 x-5=0 .$ You'll find that the roots of one equation are the opposites of the roots of the other equation. [In general, the graphs of the two equations $y=a x^{2}+b x+c$ and $y=a x^{2}-b x+c$ are symmetric about the $y$ -axis. Thus, the $x$ -intercepts (when they exist) will always be opposites of one another.

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(a) Use the quadratic formula to show that the roots of the equation $x^{2}+3 x+1=0$ are $\frac{1}{2}(-3 \pm \sqrt{5})$

(b) Show that $\frac{1}{2}(-3+\sqrt{5})=1 /\left[\frac{1}{2}(-3-\sqrt{5})\right]$

Hint: Rationalize the denominator on the right-hand side of the equation.

(c) The result in part (b) shows that the roots of the equation $x^{2}+3 x+1=0$ are reciprocals. Can you find another, much simpler way to establish this fact?

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If $r_{1}$ and $r_{2}$ are the roots of the quadratic equation $a x^{2}+b x+c=0,$ show that $r_{1}+r_{2}=-b / a$ and $r_{1} r_{2}=c / a$

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Show that the quadratic equation

$$a x^{2}+b x-a=0 \quad(a \neq 0)$$

has two distinct real roots.

Shamshad W.

Numerade Educator

Show that the quadratic equation

$$(x-p)(x-q)=r^{2} \quad(p \neq q)$$

has two distinct real roots.

Shamshad W.

Numerade Educator

Determine the value(s) of the constant $k$ for which the equation has equal roots (that is, only one distinct root.

$x^{2}=2 x(3 k+1)-7(2 k+3)$

Shamshad W.

Numerade Educator

Determine the value(s) of the constant $k$ for which the equation has equal roots (that is, only one distinct root.

$x^{2}+2(k+1) x+k^{2}=0$

Shamshad W.

Numerade Educator

Here is an outline for a slightly different derivation of the quadratic formula. The advantage of this method is that fractions are avoided until the very last step. Fill in the details.

(a) Beginning with $a x^{2}+b x=-c,$ multiply both sides by 4a. Then add $b^{2}$ to both sides.

(b) Now factor the resulting left-hand side and take square roots.

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In this section and in Section $1.3,$ we solved quadratic equations by factoring and by using the quadratic formula. This exercise shows how to solve a quadratic equation by the method of substitution. As an example, we use the equation

$$x^{2}+x-1=0$$

(a) In equation (1), make the substitution $x=y+k .$ Show that the resulting equation can be written

$$y^{2}+(2 k+1) y=1-k-k^{2}$$

(b) Find a value for $k$ so that the coefficient of $y$ in equation (2) is 0. Then, using this value of $k$, show that equation ( 2 ) becomes $y^{2}=5 / 4$

(c) Solve the equation $y^{2}=5 / 4 .$ Then use the equation $x=y+k$ to obtain the solutions of equation ( 1)

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Use the substitution method (explained in Exercise 72 ) to solve the quadratic equation $2 x^{2}-3 x+1=0$

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Assume that $a$ and $b$ are the roots of the equation $x^{2}+p x+q=0$

(a) Find the value of $a^{2} b+a b^{2}$ in terms of $p$ and $q$ Hint: Factor the expression $a^{2} b+a b^{2}$

(b) Find the value of $a^{3} b+a b^{3}$ in terms of $p$ and $q$ Hint: Factor. Then use the fact that $a^{2}+b^{2}=(a+b)^{2}-2 a b$

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@ In this exercise we investigate the effect of the constant $c$ upon the roots of the quadratic equation $x^{2}-6 x+c=0$ We do this by looking at the $x$ -intercepts of the graphs of the corresponding equations $y=x^{2}-6 x+c$

(a) Set a viewing rectangle that extends from 0 to 5 in the $x$ -direction and from $-2$ to 3 in the $y$ -direction. Then (on the same set of axes) graph the equations $y=x^{2}-6 x+c$ with c running from 8 to 10 at increments of $0.25 .$ In other words, graph the equations $y=x^{2}-6 x+8, y=x^{2}-6 x+8.25$

$y=x^{2}-6 x+8.50,$ and so on, up through $y=x^{2}-6 x+10$

(b) Note from the graphs in part (a) that, initially, as $c$ increases, the $x$ -intercepts draw closer and closer together. For which value of $c$ do the two $x$ -intercepts seem to merge into one?

(c) Use algebra as follows to check your observation in part (b). Using that value of $c$ for which there appears to be only one intercept, solve the quadratic equation $x^{2}-6 x+c=0 .$ How many roots do you obtain?

(d) Some of the graphs in part (a) have no $x$ -intercepts. What are the corresponding values of $c$ in these cases? Pick any one of these values of $c$ and use the quadratic formula to solve the equation $x^{2}-6 x+c=0 .$ What happens?

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Find nonzero real numbers $A$ and $B$ so that the roots of the equation $x^{2}+A x+B=0$ are $A$ and $B$

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