Solve x+left(x-8right)left(x+3right)=61

$$x+ \left( x-8 \right) \left( x+3 \right) = 61$$

Solve for x

$$x=\sqrt{89}+2\approx 11.433981132$$

$$x=2-\sqrt{89}\approx -7.433981132$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-8 by x+3 and combine like terms.

$$x+x^{2}-5x-24=61$$

Combine x and -5x to get -4x.

$$-4x+x^{2}-24=61$$

Subtract 61 from both sides.

$$-4x+x^{2}-24-61=0$$

Subtract 61 from -24 to get -85.

$$-4x+x^{2}-85=0$$

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.

$$x^{2}-4x-85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-85\right)}}{2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-85\right)}}{2}$$

Multiply -4 times -85.

$$x=\frac{-\left(-4\right)±\sqrt{16+340}}{2}$$

Add 16 to 340.

$$x=\frac{-\left(-4\right)±\sqrt{356}}{2}$$

Take the square root of 356.

$$x=\frac{-\left(-4\right)±2\sqrt{89}}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{89}}{2}$$

Now solve the equation x=\frac{4±2\sqrt{89}}{2} when ± is plus. Add 4 to 2\sqrt{89}.

$$x=\frac{2\sqrt{89}+4}{2}$$

Divide 4+2\sqrt{89} by 2.

$$x=\sqrt{89}+2$$

Now solve the equation x=\frac{4±2\sqrt{89}}{2} when ± is minus. Subtract 2\sqrt{89} from 4.

$$x=\frac{4-2\sqrt{89}}{2}$$

Divide 4-2\sqrt{89} by 2.

$$x=2-\sqrt{89}$$

The equation is now solved.

$$x=\sqrt{89}+2 x=2-\sqrt{89}$$

Steps for Completing the Square

Use the distributive property to multiply x-8 by x+3 and combine like terms.

$$x+x^{2}-5x-24=61$$

Combine x and -5x to get -4x.

$$-4x+x^{2}-24=61$$

Add 24 to both sides.

$$-4x+x^{2}=61+24$$

Add 61 and 24 to get 85.

$$-4x+x^{2}=85$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-4x=85$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=85+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=85+4$$

Add 85 to 4.

$$x^{2}-4x+4=89$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=89$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{89}$$

Simplify.

$$x-2=\sqrt{89} x-2=-\sqrt{89}$$

Add 2 to both sides of the equation.

$$x=\sqrt{89}+2 x=2-\sqrt{89}$$