Solve 32x-36+4x^2

$$32 x - 36 + 4 x ^ { 2 }$$

Factor

$$4\left(x-1\right)\left(x+9\right)$$

Solution Steps

Factor out 4.

$$4\left(8x-9+x^{2}\right)$$

Consider 8x-9+x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\left(-9\right)=-9$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -9.

$$-1,9 -3,3$$

Calculate the sum for each pair.

$$-1+9=8 -3+3=0$$

The solution is the pair that gives sum 8.

$$a=-1 b=9$$

Rewrite x^{2}+8x-9 as \left(x^{2}-x\right)+\left(9x-9\right).

$$\left(x^{2}-x\right)+\left(9x-9\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x-1\right)+9\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+9\right)$$

Rewrite the complete factored expression.

$$4\left(x-1\right)\left(x+9\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$4x^{2}+32x-36=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-32±\sqrt{32^{2}-4\times 4\left(-36\right)}}{2\times 4}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\times 4\left(-36\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-32±\sqrt{1024-16\left(-36\right)}}{2\times 4}$$

Multiply -16 times -36.

$$x=\frac{-32±\sqrt{1024+576}}{2\times 4}$$

Add 1024 to 576.

$$x=\frac{-32±\sqrt{1600}}{2\times 4}$$

Take the square root of 1600.

$$x=\frac{-32±40}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-32±40}{8}$$

Now solve the equation x=\frac{-32±40}{8} when ± is plus. Add -32 to 40.

$$x=\frac{8}{8}$$

Divide 8 by 8.

$$x=1$$

Now solve the equation x=\frac{-32±40}{8} when ± is minus. Subtract 40 from -32.

$$x=-\frac{72}{8}$$

Divide -72 by 8.

$$x=-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -9 for x_{2}.

$$4x^{2}+32x-36=4\left(x-1\right)\left(x-\left(-9\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$4x^{2}+32x-36=4\left(x-1\right)\left(x+9\right)$$

Evaluate

$$4\left(x-1\right)\left(x+9\right)$$