Question

question 8 of 10
which of the following are solutions to the equation below?
check all that apply.
$9x^2 - 2x + 25 = 8x^2 + 8x$

a. 6
b. -5
c. 25
d. 4
e. -3
f. 5

Explanation:

Step1: Simplify the equation

Subtract \(8x^2 + 8x\) from both sides of the equation \(9x^2 - 2x + 25 = 8x^2 + 8x\) to get:
\(9x^2 - 8x^2 - 2x - 8x + 25 = 0\)
Simplify the left - hand side: \(x^2-10x + 25 = 0\)

Step2: Factor the quadratic equation

The quadratic equation \(x^2-10x + 25 = 0\) is a perfect square trinomial. We know that \((a - b)^2=a^2-2ab + b^2\). For \(x^2-10x + 25\), \(a = x\) and \(2ab=10x\), so \(b = 5\). Then \(x^2-10x + 25=(x - 5)^2\). So the equation becomes \((x - 5)^2=0\)

Step3: Solve for x

Taking the square root of both sides of \((x - 5)^2=0\), we get \(x-5 = 0\), so \(x = 5\)

Wait, there is a mistake in the above calculation. Let's re - do the simplification step:

Step1: Simplify the equation correctly

Starting from \(9x^2-2x + 25=8x^2 + 8x\)
Subtract \(8x^2+8x\) from both sides:
\(9x^2-8x^2-2x - 8x+25 = 0\)
\(x^2-10x + 25=0\)

Step2: Factor the quadratic

We can factor \(x^2-10x + 25\) as \((x - 5)(x - 5)=0\) (since \(x^2-10x + 25=x^2-5x-5x + 25=x(x - 5)-5(x - 5)=(x - 5)(x - 5)\))

Step3: Solve for x

Set each factor equal to zero: \(x - 5=0\) or \(x - 5=0\)
So \(x = 5\)

Wait, but the options are 6, - 5,25,4, - 3,5. So the correct solution is \(x = 5\)

Wait, maybe I made a mistake in the equation simplification. Let's check again:

Original equation: \(9x^2-2x + 25=8x^2 + 8x\)

Subtract \(8x^2+8x\) from both sides:

\(9x^2-8x^2-2x-8x + 25=0\)

\(x^2-10x + 25 = 0\)

The discriminant of the quadratic equation \(ax^2+bx + c = 0\) is \(D=b^2-4ac\). For \(x^2-10x + 25\), \(a = 1\), \(b=-10\), \(c = 25\)

\(D=(-10)^2-4\times1\times25=100 - 100=0\)

Using the quadratic formula \(x=\frac{-b\pm\sqrt{D}}{2a}=\frac{10\pm\sqrt{0}}{2}=\frac{10}{2}=5\)

So the solution is \(x = 5\), which is option F.

Wait, but let's check by plugging in the values:

For option F (x = 5):

Left - hand side: \(9\times5^2-2\times5 + 25=9\times25-10 + 25=225-10 + 25=240\)

Right - hand side: \(8\times5^2+8\times5=8\times25 + 40=200 + 40=240\)

So x = 5 is a solution.

Let's check option B (x=-5):

Left - hand side: \(9\times(-5)^2-2\times(-5)+25=9\times25 + 10+25=225 + 10+25=260\)

Right - hand side: \(8\times(-5)^2+8\times(-5)=8\times25-40=200 - 40=160\)

260≠160, so x=-5 is not a solution.

Option F: x = 5, as we saw, LHS = RHS.

Wait, maybe there is a mistake in the original equation. Let me re - check the original equation. If the equation was \(9x^2-2x - 25=8x^2 + 8x\), then:

\(9x^2-8x^2-2x-8x-25 = 0\)

\(x^2-10x-25 = 0\)

Discriminant \(D = 100+100 = 200\), \(x=\frac{10\pm\sqrt{200}}{2}=\frac{10\pm10\sqrt{2}}{2}=5\pm5\sqrt{2}\), which is not in the options.

But according to the given equation \(9x^2-2x + 25=8x^2 + 8x\), the solution is x = 5, which is option F.

Answer:

F. 5