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Section 1
Quadratic Equations; Solution by Factoring
Make the given changes in the indicated examples of this section and then solve the resulting quadratic equations.In Example $3(a),$ change the $-$ sign before $7 x$ to $+$ and then solve.
Make the given changes in the indicated examples of this section and then solve the resulting quadratic equations.In Example $8,$ change the numerator of the first term to 2 and the numerator of the term on the right to 1 and then solve.
Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify $a, b,$ and $c,$ with $a>0 .$ Otherwise, explain why the resulting form is not quadratic.$$x(x-2)=4$$
Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify $a, b,$ and $c,$ with $a>0 .$ Otherwise, explain why the resulting form is not quadratic.$$(3 x-2)^{2}=2$$
Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify $a, b,$ and $c,$ with $a>0 .$ Otherwise, explain why the resulting form is not quadratic.$$x^{2}=(x+2)^{2}$$
Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify $a, b,$ and $c,$ with $a>0 .$ Otherwise, explain why the resulting form is not quadratic.$$x\left(2 x^{2}+5\right)=7+2 x^{2}$$
Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify $a, b,$ and $c,$ with $a>0 .$ Otherwise, explain why the resulting form is not quadratic.$$n\left(n^{2}+n-1\right)=n^{3}$$
Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify $a, b,$ and $c,$ with $a>0 .$ Otherwise, explain why the resulting form is not quadratic.$$(T-7)^{2}=(2 T+3)^{2}$$
Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify $a, b,$ and $c,$ with $a>0 .$ Otherwise, explain why the resulting form is not quadratic.$$y^{2}(y-2)=3(y-2)$$
Determine whether or not the given equations are quadratic. If the resulting form is quadratic, identify $a, b,$ and $c,$ with $a>0 .$ Otherwise, explain why the resulting form is not quadratic.$$z(z+4)=(z+1)(z+5)$$
Solve the given quadratic equations by factoring.$$x^{2}-25=0$$
Solve the given quadratic equations by factoring.$$B^{2}-400=0$$
Solve the given quadratic equations by factoring.$$4 y^{2}=9$$
Solve the given quadratic equations by factoring.$$2 x^{2}=0.32$$
Solve the given quadratic equations by factoring.$$x^{2}-5 x-14=0$$
Solve the given quadratic equations by factoring.$$x^{2}+x-6=0$$
Solve the given quadratic equations by factoring.$$R^{2}+12=7 R$$
Solve the given quadratic equations by factoring.$$x^{2}+30=11 x$$
Solve the given quadratic equations by factoring.$$40 x-16 x^{2}=0$$
Solve the given quadratic equations by factoring.$$15 L=20 L^{2}$$
Solve the given quadratic equations by factoring.$$12 m^{2}=3$$
Solve the given quadratic equations by factoring.$$9=a^{2} x^{2}$$
Solve the given quadratic equations by factoring.$$3 x^{2}-13 x+4=0$$
Solve the given quadratic equations by factoring.$$A^{2}+8 A+16=0$$
Solve the given quadratic equations by factoring.$$4 x=3-7 x^{2}$$
Solve the given quadratic equations by factoring.$$4 x^{2}+25=20 x$$
Solve the given quadratic equations by factoring.$$6 x^{2}=13 x-6$$
Solve the given quadratic equations by factoring.$$6 z^{2}=6+5 z$$
Solve the given quadratic equations by factoring.$$4 x(x+1)=3$$
Solve the given quadratic equations by factoring.$$t(43+t)=9-9 t^{2}$$
Solve the given quadratic equations by factoring.$$6 y^{2}+b y=2 b^{2}$$
Solve the given quadratic equations by factoring.$$2 x^{2}-7 a x+4 a^{2}=a^{2}$$
Solve the given quadratic equations by factoring.$$8 s^{2}+16 s=90$$
Solve the given quadratic equations by factoring.$$18 t^{2}=48 t-32$$
Solve the given quadratic equations by factoring.$$(x+2)^{3}=x^{3}+8$$
Solve the given quadratic equations by factoring.$$V\left(V^{2}-4\right)=V^{2}(V-1)$$
Solve the given quadratic equations by factoring.$$(x+a)^{2}-b^{2}=0$$
Solve the given quadratic equations by factoring.$$b x^{2}-b=x-b^{2} x$$
Solve the given quadratic equations by factoring.$$x^{2}+2 a x=b^{2}-a^{2}$$
Solve the given quadratic equations by factoring.$$x^{2}\left(a^{2}+2 a b+b^{2}\right)=x(a+b)$$
In Eq. $(7.1),$ for $a=2, b=-7,$ and $c=3,$ show that the sum of the roots is $-b / a$.
For the equation of Exercise 41 show that the product of the roots is $c / a$.
In finding the dimensions of a crate, the equation $12 x^{2}-64 x+64=0$ is used. Solve for $x,$ if $x>2$.
If a rocket is launched with an initial velocity of $320 \mathrm{ft} / \mathrm{s}$, its height above ground after $t$ seconds is given by $-16 t^{2}+320 t$ (in ft). Find the times when the height is $0 .$
The voltage $V$ across a semiconductor in a computer is given by $V=\alpha I+\beta I^{2},$ where $I$ is the current (in A). If a 6 -V battery is conducted across the semiconductor, find the current if $\alpha=2 \Omega$ and $\beta=0.5 \Omega / \mathrm{A}$.
The mass $m$ (in $\mathrm{Mg}$ ) of the fuel supply in the first-stage booster of a rocket is $m=135-6 t-t^{2},$ where $t$ is the time (in s) after launch. When does the booster run out of fuel?
The power $P$ (in $\mathrm{MW}$ ) produced between midnight and noon by a nuclear power plant is $P=4 h^{2}-48 h+744,$ where $h$ is the hour of the day. At what time is the power 664 MW?
In determining the speed $s$ (in mi/h) of a car while studying its fuel economy, the equation $s^{2}-16 s=3072$ is used. Find $s$.
Find the indicated quadratic equations.Find a quadratic equation for which the solutions are 0.5 and 2.
Find the indicated quadratic equations.Find a quadratic equation for which the solutions are $a$ and $b$.
Although the equations are not quadratic, factoring will lead to one quadratic factor and the solution can be completed by factoring as with a quadratic equation. Find the three roots of each equation.$$x^{3}-x=0$$
Although the equations are not quadratic, factoring will lead to one quadratic factor and the solution can be completed by factoring as with a quadratic equation. Find the three roots of each equation.$$x^{3}-4 x^{2}-x+4=0$$
Solve the given equations involving fractions.$$\frac{1}{x-3}+\frac{4}{x}=2$$
Solve the given equations involving fractions.$$2-\frac{1}{x}=\frac{3}{x+2}$$
Solve the given equations involving fractions.$$\frac{1}{2 x}-\frac{3}{4}=\frac{1}{2 x+3}$$
Solve the given equations involving fractions.$$\frac{x}{2}+\frac{1}{x-3}=3$$
Set up the appropriate quadratic equations and solve.The spring constant $k$ is the force $F$ divided by the amount $x$ the spring stretches $(k=F / x) .$ See Fig. $7.2(a)$ For two springs in series [see Fig. $7.2(\mathrm{b})],$ the reciprocal of the spring constant $k_{c}$ for the combination equals the sum of the reciprocals of the individual spring constants. Find the spring constants for each of two springs in series if $k_{c}=2 \mathrm{N} / \mathrm{cm}$ and one spring constant is $3 \mathrm{N} / \mathrm{cm}$ more than the other.
Set up the appropriate quadratic equations and solve.The combined resistance $R$ of two resistances $R_{1}$ and $R_{2}$ connected in parallel [see Fig. $7.3(\text { a) }]$ is equal to the product of the individual resistances divided by their sum. If the two resistances are connected in series [see Fig. $7.3(\mathrm{b})]$, their combined resistance is the sum of their individual resistances. If two resistances connected in parallel have a combined resistance of $3.0 \Omega$ and the same two resistances have a combined resistance of $16 \Omega$ when connected in series, what are the resistances?
A hydrofoil made the round-trip of $120 \mathrm{km}$ between two islands in$3.5 \mathrm{h}$ of travel time. If the average speed going was $10 \mathrm{km} / \mathrm{h}$ less than the average speed returning, find these speeds.
Set up the appropriate quadratic equations and solve.A rectangular solar panel is $20 \mathrm{cm}$ by $30 \mathrm{cm} .$ By adding the same amount to each dimension, the area is doubled. How much is added?