Solve 7x=28+x(71-x)

$$7x=28+x(71-x)$$

Solve for x

$$x=2\sqrt{263}+32\approx 64.43454948$$

$$x=32-2\sqrt{263}\approx -0.43454948$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 71-x.

$$7x=28+71x-x^{2}$$

Subtract 28 from both sides.

$$7x-28=71x-x^{2}$$

Subtract 71x from both sides.

$$7x-28-71x=-x^{2}$$

Combine 7x and -71x to get -64x.

$$-64x-28=-x^{2}$$

Add x^{2} to both sides.

$$-64x-28+x^{2}=0$$

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.

$$x^{2}-64x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -64 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\left(-28\right)}}{2}$$

Square -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\left(-28\right)}}{2}$$

Multiply -4 times -28.

$$x=\frac{-\left(-64\right)±\sqrt{4096+112}}{2}$$

Add 4096 to 112.

$$x=\frac{-\left(-64\right)±\sqrt{4208}}{2}$$

Take the square root of 4208.

$$x=\frac{-\left(-64\right)±4\sqrt{263}}{2}$$

The opposite of -64 is 64.

$$x=\frac{64±4\sqrt{263}}{2}$$

Now solve the equation x=\frac{64±4\sqrt{263}}{2} when ± is plus. Add 64 to 4\sqrt{263}.

$$x=\frac{4\sqrt{263}+64}{2}$$

Divide 64+4\sqrt{263} by 2.

$$x=2\sqrt{263}+32$$

Now solve the equation x=\frac{64±4\sqrt{263}}{2} when ± is minus. Subtract 4\sqrt{263} from 64.

$$x=\frac{64-4\sqrt{263}}{2}$$

Divide 64-4\sqrt{263} by 2.

$$x=32-2\sqrt{263}$$

The equation is now solved.

$$x=2\sqrt{263}+32 x=32-2\sqrt{263}$$

Steps for Completing the Square

Use the distributive property to multiply x by 71-x.

$$7x=28+71x-x^{2}$$

Subtract 71x from both sides.

$$7x-71x=28-x^{2}$$

Combine 7x and -71x to get -64x.

$$-64x=28-x^{2}$$

Add x^{2} to both sides.

$$-64x+x^{2}=28$$

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.

$$x^{2}-64x=28$$

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

$$x^{2}-64x+\left(-32\right)^{2}=28+\left(-32\right)^{2}$$

Square -32.

$$x^{2}-64x+1024=28+1024$$

Add 28 to 1024.

$$x^{2}-64x+1024=1052$$

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

$$\left(x-32\right)^{2}=1052$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-32\right)^{2}}=\sqrt{1052}$$

Simplify.

$$x-32=2\sqrt{263} x-32=-2\sqrt{263}$$

Add 32 to both sides of the equation.

$$x=2\sqrt{263}+32 x=32-2\sqrt{263}$$