Question

find the x - intercepts of the graph of the quadratic function. then match the function with its graph.
y = x^{2}-10x + 24
the x - intercepts are
(type integers or fractions. use a comma to separate answers as needed.)
choose the correct graph below. each graph is shown in a -10,10,1 by -10,10,1 viewing rectangle.

1. solve the equation by the method of your choice.

6x^{2}-3x = 2
the solution set is
(simplify your answer. type integers or fractions. use a comma to separate answers.)

1. solve the following equation by the method of your choice. simplify irrational solutions, if possible.

5x^{2}-13x=-6
the solution set is
(simplify your answer. use a comma to separate answers as needed.)

1. solve.

(7x + 4)^{2}=48
the solution set is
(type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)

1. graph the parabola whose equation is given.

y = x^{2}+10x + 9
graph the parabola using the tool to the right.

Explanation:

Step1: Solve \(6x^{2}-3x = 2\)

Rewrite the equation as \(6x^{2}-3x - 2=0\).
Use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a = 6\), \(b=-3\), \(c=-2\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-3)^{2}-4\times6\times(-2)=9 + 48=57\).
Then \(x=\frac{3\pm\sqrt{57}}{12}\).

Step2: Solve \(5x^{2}-13x=-6\)

Rewrite the equation as \(5x^{2}-13x + 6=0\).
Factor the quadratic: \(5x^{2}-13x + 6=(5x - 3)(x - 2)=0\).
Set each factor equal to zero: \(5x-3 = 0\) gives \(x=\frac{3}{5}\), and \(x - 2=0\) gives \(x = 2\).

Step3: Solve \((7x + 4)^{2}=48\)

Take the square root of both sides: \(7x+4=\pm\sqrt{48}=\pm4\sqrt{3}\).
Then \(7x=-4\pm4\sqrt{3}\), and \(x=\frac{-4\pm4\sqrt{3}}{7}\).

Answer:

1. \(\frac{3+\sqrt{57}}{12},\frac{3-\sqrt{57}}{12}\)
2. \(2,\frac{3}{5}\)
3. \(\frac{-4 + 4\sqrt{3}}{7},\frac{-4-4\sqrt{3}}{7}\)