Solve 0=7x^2+16x-15

$$0 = 7 x ^ { 2 } + 16 x - 15$$

Solve for x

$$x=-3$$

$$x=\frac{5}{7}\approx 0.714285714$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$7x^{2}+16x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=16 ab=7\left(-15\right)=-105$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -105.

$$-1,105 -3,35 -5,21 -7,15$$

Calculate the sum for each pair.

$$-1+105=104 -3+35=32 -5+21=16 -7+15=8$$

The solution is the pair that gives sum 16.

$$a=-5 b=21$$

Rewrite 7x^{2}+16x-15 as \left(7x^{2}-5x\right)+\left(21x-15\right).

$$\left(7x^{2}-5x\right)+\left(21x-15\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(7x-5\right)+3\left(7x-5\right)$$

Factor out common term 7x-5 by using distributive property.

$$\left(7x-5\right)\left(x+3\right)$$

To find equation solutions, solve 7x-5=0 and x+3=0.

$$x=\frac{5}{7} x=-3$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$7x^{2}+16x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 16 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 7\left(-15\right)}}{2\times 7}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 7\left(-15\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-16±\sqrt{256-28\left(-15\right)}}{2\times 7}$$

Multiply -28 times -15.

$$x=\frac{-16±\sqrt{256+420}}{2\times 7}$$

Add 256 to 420.

$$x=\frac{-16±\sqrt{676}}{2\times 7}$$

Take the square root of 676.

$$x=\frac{-16±26}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-16±26}{14}$$

Now solve the equation x=\frac{-16±26}{14} when ± is plus. Add -16 to 26.

$$x=\frac{10}{14}$$

Reduce the fraction \frac{10}{14} to lowest terms by extracting and canceling out 2.

$$x=\frac{5}{7}$$

Now solve the equation x=\frac{-16±26}{14} when ± is minus. Subtract 26 from -16.

$$x=-\frac{42}{14}$$

Divide -42 by 14.

$$x=-3$$

The equation is now solved.

$$x=\frac{5}{7} x=-3$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$7x^{2}+16x-15=0$$

Add 15 to both sides. Anything plus zero gives itself.

$$7x^{2}+16x=15$$

Divide both sides by 7.

$$\frac{7x^{2}+16x}{7}=\frac{15}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{16}{7}x=\frac{15}{7}$$

Divide \frac{16}{7}, the coefficient of the x term, by 2 to get \frac{8}{7}. Then add the square of \frac{8}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{7}x+\left(\frac{8}{7}\right)^{2}=\frac{15}{7}+\left(\frac{8}{7}\right)^{2}$$

Square \frac{8}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{7}x+\frac{64}{49}=\frac{15}{7}+\frac{64}{49}$$

Add \frac{15}{7} to \frac{64}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{7}x+\frac{64}{49}=\frac{169}{49}$$

Factor x^{2}+\frac{16}{7}x+\frac{64}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{7}\right)^{2}=\frac{169}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{7}\right)^{2}}=\sqrt{\frac{169}{49}}$$

Simplify.

$$x+\frac{8}{7}=\frac{13}{7} x+\frac{8}{7}=-\frac{13}{7}$$

Subtract \frac{8}{7} from both sides of the equation.

$$x=\frac{5}{7} x=-3$$