Solve (-8+x(64+43x))-1=0

$$(-8+x(64+43x))-1=0$$

Solve for x

$$x=\frac{\sqrt{1411}-32}{43}\approx 0.129378603$$

$$x=\frac{-\sqrt{1411}-32}{43}\approx -1.617750696$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 64+43x.

$$-8+64x+43x^{2}-1=0$$

Subtract 1 from -8 to get -9.

$$-9+64x+43x^{2}=0$$

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.

$$43x^{2}+64x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 43 for a, 64 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-64±\sqrt{64^{2}-4\times 43\left(-9\right)}}{2\times 43}$$

Square 64.

$$x=\frac{-64±\sqrt{4096-4\times 43\left(-9\right)}}{2\times 43}$$

Multiply -4 times 43.

$$x=\frac{-64±\sqrt{4096-172\left(-9\right)}}{2\times 43}$$

Multiply -172 times -9.

$$x=\frac{-64±\sqrt{4096+1548}}{2\times 43}$$

Add 4096 to 1548.

$$x=\frac{-64±\sqrt{5644}}{2\times 43}$$

Take the square root of 5644.

$$x=\frac{-64±2\sqrt{1411}}{2\times 43}$$

Multiply 2 times 43.

$$x=\frac{-64±2\sqrt{1411}}{86}$$

Now solve the equation x=\frac{-64±2\sqrt{1411}}{86} when ± is plus. Add -64 to 2\sqrt{1411}.

$$x=\frac{2\sqrt{1411}-64}{86}$$

Divide -64+2\sqrt{1411} by 86.

$$x=\frac{\sqrt{1411}-32}{43}$$

Now solve the equation x=\frac{-64±2\sqrt{1411}}{86} when ± is minus. Subtract 2\sqrt{1411} from -64.

$$x=\frac{-2\sqrt{1411}-64}{86}$$

Divide -64-2\sqrt{1411} by 86.

$$x=\frac{-\sqrt{1411}-32}{43}$$

The equation is now solved.

$$x=\frac{\sqrt{1411}-32}{43} x=\frac{-\sqrt{1411}-32}{43}$$

Steps for Completing the Square

Use the distributive property to multiply x by 64+43x.

$$-8+64x+43x^{2}-1=0$$

Subtract 1 from -8 to get -9.

$$-9+64x+43x^{2}=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$64x+43x^{2}=9$$

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.

$$43x^{2}+64x=9$$

Divide both sides by 43.

$$\frac{43x^{2}+64x}{43}=\frac{9}{43}$$

Dividing by 43 undoes the multiplication by 43.

$$x^{2}+\frac{64}{43}x=\frac{9}{43}$$

Divide \frac{64}{43}, the coefficient of the x term, by 2 to get \frac{32}{43}. Then add the square of \frac{32}{43} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{64}{43}x+\left(\frac{32}{43}\right)^{2}=\frac{9}{43}+\left(\frac{32}{43}\right)^{2}$$

Square \frac{32}{43} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{64}{43}x+\frac{1024}{1849}=\frac{9}{43}+\frac{1024}{1849}$$

Add \frac{9}{43} to \frac{1024}{1849} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{64}{43}x+\frac{1024}{1849}=\frac{1411}{1849}$$

Factor x^{2}+\frac{64}{43}x+\frac{1024}{1849}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{32}{43}\right)^{2}=\frac{1411}{1849}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{32}{43}\right)^{2}}=\sqrt{\frac{1411}{1849}}$$

Simplify.

$$x+\frac{32}{43}=\frac{\sqrt{1411}}{43} x+\frac{32}{43}=-\frac{\sqrt{1411}}{43}$$

Subtract \frac{32}{43} from both sides of the equation.

$$x=\frac{\sqrt{1411}-32}{43} x=\frac{-\sqrt{1411}-32}{43}$$