Solve (2x-1)(x-8)-(x-1)(x+1)=15x

$$(2x-1)(x-8)-(x-1)(x+1)=15x$$

Solve for x

$$x=\sqrt{247}+16\approx 31.716233646$$

$$x=16-\sqrt{247}\approx 0.283766354$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x-1 by x-8 and combine like terms.

$$2x^{2}-17x+8-\left(x-1\right)\left(x+1\right)=15x$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$2x^{2}-17x+8-\left(x^{2}-1\right)=15x$$

To find the opposite of x^{2}-1, find the opposite of each term.

$$2x^{2}-17x+8-x^{2}+1=15x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-17x+8+1=15x$$

Add 8 and 1 to get 9.

$$x^{2}-17x+9=15x$$

Subtract 15x from both sides.

$$x^{2}-17x+9-15x=0$$

Combine -17x and -15x to get -32x.

$$x^{2}-32x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -32 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 9}}{2}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 9}}{2}$$

Multiply -4 times 9.

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

Add 1024 to -36.

$$x=\frac{-\left(-32\right)±\sqrt{988}}{2}$$

Take the square root of 988.

$$x=\frac{-\left(-32\right)±2\sqrt{247}}{2}$$

The opposite of -32 is 32.

$$x=\frac{32±2\sqrt{247}}{2}$$

Now solve the equation x=\frac{32±2\sqrt{247}}{2} when ± is plus. Add 32 to 2\sqrt{247}.

$$x=\frac{2\sqrt{247}+32}{2}$$

Divide 32+2\sqrt{247} by 2.

$$x=\sqrt{247}+16$$

Now solve the equation x=\frac{32±2\sqrt{247}}{2} when ± is minus. Subtract 2\sqrt{247} from 32.

$$x=\frac{32-2\sqrt{247}}{2}$$

Divide 32-2\sqrt{247} by 2.

$$x=16-\sqrt{247}$$

The equation is now solved.

$$x=\sqrt{247}+16 x=16-\sqrt{247}$$

Steps for Completing the Square

Use the distributive property to multiply 2x-1 by x-8 and combine like terms.

$$2x^{2}-17x+8-\left(x-1\right)\left(x+1\right)=15x$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

$$2x^{2}-17x+8-\left(x^{2}-1\right)=15x$$

To find the opposite of x^{2}-1, find the opposite of each term.

$$2x^{2}-17x+8-x^{2}+1=15x$$

Combine 2x^{2} and -x^{2} to get x^{2}.

$$x^{2}-17x+8+1=15x$$

Add 8 and 1 to get 9.

$$x^{2}-17x+9=15x$$

Subtract 15x from both sides.

$$x^{2}-17x+9-15x=0$$

Combine -17x and -15x to get -32x.

$$x^{2}-32x+9=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-32x=-9$$

Divide -32, the coefficient of the x term, by 2 to get -16. Then add the square of -16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-32x+\left(-16\right)^{2}=-9+\left(-16\right)^{2}$$

Square -16.

$$x^{2}-32x+256=-9+256$$

Add -9 to 256.

$$x^{2}-32x+256=247$$

Factor x^{2}-32x+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-16\right)^{2}=247$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-16\right)^{2}}=\sqrt{247}$$

Simplify.

$$x-16=\sqrt{247} x-16=-\sqrt{247}$$

Add 16 to both sides of the equation.

$$x=\sqrt{247}+16 x=16-\sqrt{247}$$