Solve 7(-x+5)=9x^2+9x

$$7 ( - x + 5 ) = 9 x ^ { 2 } + 9 x$$

Solve for x

$$x = \frac{\sqrt{379} - 8}{9} \approx 1.274213593$$

$$x=\frac{-\sqrt{379}-8}{9}\approx -3.05199137$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 7 by -x+5.

$$7\left(-x\right)+35=9x^{2}+9x$$

Subtract 9x^{2} from both sides.

$$7\left(-x\right)+35-9x^{2}=9x$$

Subtract 9x from both sides.

$$7\left(-x\right)+35-9x^{2}-9x=0$$

Multiply 7 and -1 to get -7.

$$-7x+35-9x^{2}-9x=0$$

Combine -7x and -9x to get -16x.

$$-16x+35-9x^{2}=0$$

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.

$$-9x^{2}-16x+35=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -16 for b, and 35 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-9\right)\times 35}}{2\left(-9\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-9\right)\times 35}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-16\right)±\sqrt{256+36\times 35}}{2\left(-9\right)}$$

Multiply 36 times 35.

$$x=\frac{-\left(-16\right)±\sqrt{256+1260}}{2\left(-9\right)}$$

Add 256 to 1260.

$$x=\frac{-\left(-16\right)±\sqrt{1516}}{2\left(-9\right)}$$

Take the square root of 1516.

$$x=\frac{-\left(-16\right)±2\sqrt{379}}{2\left(-9\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{379}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{16±2\sqrt{379}}{-18}$$

Now solve the equation x=\frac{16±2\sqrt{379}}{-18} when ± is plus. Add 16 to 2\sqrt{379}.

$$x=\frac{2\sqrt{379}+16}{-18}$$

Divide 16+2\sqrt{379} by -18.

$$x=\frac{-\sqrt{379}-8}{9}$$

Now solve the equation x=\frac{16±2\sqrt{379}}{-18} when ± is minus. Subtract 2\sqrt{379} from 16.

$$x=\frac{16-2\sqrt{379}}{-18}$$

Divide 16-2\sqrt{379} by -18.

$$x=\frac{\sqrt{379}-8}{9}$$

The equation is now solved.

$$x=\frac{-\sqrt{379}-8}{9} x=\frac{\sqrt{379}-8}{9}$$

Steps for Completing the Square

Use the distributive property to multiply 7 by -x+5.

$$7\left(-x\right)+35=9x^{2}+9x$$

Subtract 9x^{2} from both sides.

$$7\left(-x\right)+35-9x^{2}=9x$$

Subtract 9x from both sides.

$$7\left(-x\right)+35-9x^{2}-9x=0$$

Subtract 35 from both sides. Anything subtracted from zero gives its negation.

$$7\left(-x\right)-9x^{2}-9x=-35$$

Multiply 7 and -1 to get -7.

$$-7x-9x^{2}-9x=-35$$

Combine -7x and -9x to get -16x.

$$-16x-9x^{2}=-35$$

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.

$$-9x^{2}-16x=-35$$

Divide both sides by -9.

$$\frac{-9x^{2}-16x}{-9}=-\frac{35}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{16}{-9}\right)x=-\frac{35}{-9}$$

Divide -16 by -9.

$$x^{2}+\frac{16}{9}x=-\frac{35}{-9}$$

Divide -35 by -9.

$$x^{2}+\frac{16}{9}x=\frac{35}{9}$$

Divide \frac{16}{9}, the coefficient of the x term, by 2 to get \frac{8}{9}. Then add the square of \frac{8}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{9}x+\left(\frac{8}{9}\right)^{2}=\frac{35}{9}+\left(\frac{8}{9}\right)^{2}$$

Square \frac{8}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{9}x+\frac{64}{81}=\frac{35}{9}+\frac{64}{81}$$

Add \frac{35}{9} to \frac{64}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{9}x+\frac{64}{81}=\frac{379}{81}$$

Factor x^{2}+\frac{16}{9}x+\frac{64}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{9}\right)^{2}=\frac{379}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{9}\right)^{2}}=\sqrt{\frac{379}{81}}$$

Simplify.

$$x+\frac{8}{9}=\frac{\sqrt{379}}{9} x+\frac{8}{9}=-\frac{\sqrt{379}}{9}$$

Subtract \frac{8}{9} from both sides of the equation.

$$x=\frac{\sqrt{379}-8}{9} x=\frac{-\sqrt{379}-8}{9}$$