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Section 1
Real Roots of a Quadratic Equation
To solve the equation given in step 2 of Example 2 , Sarah multiplied each side of the equation by 8 and then added 1 to each side to complete the square. Show that $16 x^{2}-8 x-8$ is the square of a binomial and will lead to the correct solution of $0=2 x^{2}-x-1$
Phillip said that the equation $0=x^{2}-6 x+1$ can be solved by adding 8 to both sides of the equation. Do you agree with Phillip? Explain why or why not.
In $3-8,$ complete the square of the quadratic expression.$$x^{2}+6 x$$
In $3-8,$ complete the square of the quadratic expression.$$x^{2}-8 x$$
In $3-8,$ complete the square of the quadratic expression.$$x^{2}-2 x$$
In $3-8,$ complete the square of the quadratic expression.$$x^{2}-12 x$$
In $3-8,$ complete the square of the quadratic expression.$$2 x^{2}-4 x$$
In $3-8,$ complete the square of the quadratic expression.$$x^{2}-3 x$$
In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.$$\mathrm{f}(x)=x^{2}-6 x+4$$
In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.$$\mathrm{f}(x)=x^{2}-2 x-2$$
In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.$$f(x)=x^{2}+4 x+2$$
In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.$$f(x)=x^{2}-6 x+6$$
In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.$$f(x)=x^{2}-2 x-1$$
In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.$$f(x)=x^{2}-10 x+18$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$x^{2}-2 x-2=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$x^{2}+6 x+4=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$x^{2}-4 x+1=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$x^{2}+2 x-5=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$x^{2}-6 x+2=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$x^{2}-8 x+4=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$2 x^{2}+12 x+3=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$2 x^{2}-6 x+3=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$4 x^{2}-20 x+9=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$\frac{1}{2} x^{2}+x-3=0$$
In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.$$\frac{1}{4} x^{2}+\frac{3}{4} x-\frac{3}{2}=0$$
a. Complete the square to find the roots of the equation $x^{2}-5 x+1=0$ .b. Write, to the nearest tenth, a rational approximation for the roots.
In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.$$\mathrm{f}(x)=x^{2}-12 x+5$$
In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.$$\mathrm{f}(x)=x^{2}+2 x-2$$
In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.$$f(x)=x^{2}-6 x-7$$
In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.$$\mathrm{f}(x)=x^{2}+x+\frac{9}{4}$$
In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.$$\mathrm{f}(x)=-x^{2}+x+2$$
In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.$$f(x)=3 x^{2}+6 x+3$$
Determine the coordinates of the vertex and the equation of the axis of symmetry of $\mathrm{f}(x)=x^{2}+8 x+5$ by writing the equation in the form $\mathrm{f}(x)=(x-h)^{2}+k .$ Justify your answer.
The length of a rectangle is 4 feet more than twice the width. The area of the rectangle is 38 square feet.a. Find the dimensions of the rectangle in simplest radical form.b. Show that the product of the length and width is equal to the area.c. Write, to the nearest tenth, rational approximations for the length and width.
One base of a trapezoid is 8 feet longer than the other base and the height of the trapezoid is equal to the length of the shorter base. The area of the trapezoid is 20 square feet.a. Find the lengths of the bases and of the height of the trapezoid in simplest radical form.b. Show the lengths of the basezoid is equal to one-half the height times the sum of the lengths of the bases.c. Write, to the nearest tenth, rational approximations for the lengths of the bases and for the height of the trapezoid.
Steve and Alice realized that their ages are consecutive odd integers. The product of their ages is $195 .$ Steve is younger than Alice. Determine their ages by completing the square.