Solve 4x*(4x-21)=80

$$4 x \cdot ( 4 x - 21 ) = 80$$

Solve for x

$$x = \frac{\sqrt{761} + 21}{8} \approx 6.073278556$$

$$x=\frac{21-\sqrt{761}}{8}\approx -0.823278556$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by 4x-21.

$$16x^{2}-84x=80$$

Subtract 80 from both sides.

$$16x^{2}-84x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -84 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-84\right)±\sqrt{\left(-84\right)^{2}-4\times 16\left(-80\right)}}{2\times 16}$$

Square -84.

$$x=\frac{-\left(-84\right)±\sqrt{7056-4\times 16\left(-80\right)}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-84\right)±\sqrt{7056-64\left(-80\right)}}{2\times 16}$$

Multiply -64 times -80.

$$x=\frac{-\left(-84\right)±\sqrt{7056+5120}}{2\times 16}$$

Add 7056 to 5120.

$$x=\frac{-\left(-84\right)±\sqrt{12176}}{2\times 16}$$

Take the square root of 12176.

$$x=\frac{-\left(-84\right)±4\sqrt{761}}{2\times 16}$$

The opposite of -84 is 84.

$$x=\frac{84±4\sqrt{761}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{84±4\sqrt{761}}{32}$$

Now solve the equation x=\frac{84±4\sqrt{761}}{32} when ± is plus. Add 84 to 4\sqrt{761}.

$$x=\frac{4\sqrt{761}+84}{32}$$

Divide 84+4\sqrt{761} by 32.

$$x=\frac{\sqrt{761}+21}{8}$$

Now solve the equation x=\frac{84±4\sqrt{761}}{32} when ± is minus. Subtract 4\sqrt{761} from 84.

$$x=\frac{84-4\sqrt{761}}{32}$$

Divide 84-4\sqrt{761} by 32.

$$x=\frac{21-\sqrt{761}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{761}+21}{8} x=\frac{21-\sqrt{761}}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 4x by 4x-21.

$$16x^{2}-84x=80$$

Divide both sides by 16.

$$\frac{16x^{2}-84x}{16}=\frac{80}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\left(-\frac{84}{16}\right)x=\frac{80}{16}$$

Reduce the fraction \frac{-84}{16} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{21}{4}x=\frac{80}{16}$$

Divide 80 by 16.

$$x^{2}-\frac{21}{4}x=5$$

Divide -\frac{21}{4}, the coefficient of the x term, by 2 to get -\frac{21}{8}. Then add the square of -\frac{21}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{21}{4}x+\left(-\frac{21}{8}\right)^{2}=5+\left(-\frac{21}{8}\right)^{2}$$

Square -\frac{21}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{21}{4}x+\frac{441}{64}=5+\frac{441}{64}$$

Add 5 to \frac{441}{64}.

$$x^{2}-\frac{21}{4}x+\frac{441}{64}=\frac{761}{64}$$

Factor x^{2}-\frac{21}{4}x+\frac{441}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{8}\right)^{2}=\frac{761}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{8}\right)^{2}}=\sqrt{\frac{761}{64}}$$

Simplify.

$$x-\frac{21}{8}=\frac{\sqrt{761}}{8} x-\frac{21}{8}=-\frac{\sqrt{761}}{8}$$

Add \frac{21}{8} to both sides of the equation.

$$x=\frac{\sqrt{761}+21}{8} x=\frac{21-\sqrt{761}}{8}$$