Solve 4x(x-3)-(7-x)=80

$$4 x ( x - 3 ) - ( 7 - x ) = 80$$

Solve for x

$$x = \frac{\sqrt{1513} + 11}{8} \approx 6.237162585$$

$$x=\frac{11-\sqrt{1513}}{8}\approx -3.487162585$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by x-3.

$$4x^{2}-12x-\left(7-x\right)=80$$

To find the opposite of 7-x, find the opposite of each term.

$$4x^{2}-12x-7-\left(-x\right)=80$$

The opposite of -x is x.

$$4x^{2}-12x-7+x=80$$

Combine -12x and x to get -11x.

$$4x^{2}-11x-7=80$$

Subtract 80 from both sides.

$$4x^{2}-11x-7-80=0$$

Subtract 80 from -7 to get -87.

$$4x^{2}-11x-87=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -11 for b, and -87 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 4\left(-87\right)}}{2\times 4}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 4\left(-87\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-11\right)±\sqrt{121-16\left(-87\right)}}{2\times 4}$$

Multiply -16 times -87.

$$x=\frac{-\left(-11\right)±\sqrt{121+1392}}{2\times 4}$$

Add 121 to 1392.

$$x=\frac{-\left(-11\right)±\sqrt{1513}}{2\times 4}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{1513}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{11±\sqrt{1513}}{8}$$

Now solve the equation x=\frac{11±\sqrt{1513}}{8} when ± is plus. Add 11 to \sqrt{1513}.

$$x=\frac{\sqrt{1513}+11}{8}$$

Now solve the equation x=\frac{11±\sqrt{1513}}{8} when ± is minus. Subtract \sqrt{1513} from 11.

$$x=\frac{11-\sqrt{1513}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{1513}+11}{8} x=\frac{11-\sqrt{1513}}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 4x by x-3.

$$4x^{2}-12x-\left(7-x\right)=80$$

To find the opposite of 7-x, find the opposite of each term.

$$4x^{2}-12x-7-\left(-x\right)=80$$

The opposite of -x is x.

$$4x^{2}-12x-7+x=80$$

Combine -12x and x to get -11x.

$$4x^{2}-11x-7=80$$

Add 7 to both sides.

$$4x^{2}-11x=80+7$$

Add 80 and 7 to get 87.

$$4x^{2}-11x=87$$

Divide both sides by 4.

$$\frac{4x^{2}-11x}{4}=\frac{87}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{11}{4}x=\frac{87}{4}$$

Divide -\frac{11}{4}, the coefficient of the x term, by 2 to get -\frac{11}{8}. Then add the square of -\frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=\frac{87}{4}+\left(-\frac{11}{8}\right)^{2}$$

Square -\frac{11}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=\frac{87}{4}+\frac{121}{64}$$

Add \frac{87}{4} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=\frac{1513}{64}$$

Factor x^{2}-\frac{11}{4}x+\frac{121}{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}.

$$\left(x-\frac{11}{8}\right)^{2}=\frac{1513}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{8}\right)^{2}}=\sqrt{\frac{1513}{64}}$$

Simplify.

$$x-\frac{11}{8}=\frac{\sqrt{1513}}{8} x-\frac{11}{8}=-\frac{\sqrt{1513}}{8}$$

Add \frac{11}{8} to both sides of the equation.

$$x=\frac{\sqrt{1513}+11}{8} x=\frac{11-\sqrt{1513}}{8}$$