Solve 0=x^2+17x+70

$$0 = x ^ { 2 } + 17 x + 70$$

Solve for x

$$x=-10$$

$$x=-7$$

Steps Using Factoring

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

$$x^{2}+17x+70=0$$

To solve the equation, factor x^{2}+17x+70 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=17 ab=70$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 70.

$$1,70 2,35 5,14 7,10$$

Calculate the sum for each pair.

$$1+70=71 2+35=37 5+14=19 7+10=17$$

The solution is the pair that gives sum 17.

$$a=7 b=10$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x+7\right)\left(x+10\right)$$

To find equation solutions, solve x+7=0 and x+10=0.

$$x=-7 x=-10$$

Steps Using Factoring By Grouping

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

$$x^{2}+17x+70=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+70. To find a and b, set up a system to be solved.

$$a+b=17 ab=1\times 70=70$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 70.

$$1,70 2,35 5,14 7,10$$

Calculate the sum for each pair.

$$1+70=71 2+35=37 5+14=19 7+10=17$$

The solution is the pair that gives sum 17.

$$a=7 b=10$$

Rewrite x^{2}+17x+70 as \left(x^{2}+7x\right)+\left(10x+70\right).

$$\left(x^{2}+7x\right)+\left(10x+70\right)$$

Factor out x in the first and 10 in the second group.

$$x\left(x+7\right)+10\left(x+7\right)$$

Factor out common term x+7 by using distributive property.

$$\left(x+7\right)\left(x+10\right)$$

To find equation solutions, solve x+7=0 and x+10=0.

$$x=-7 x=-10$$

Steps Using the Quadratic Formula

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

$$x^{2}+17x+70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 17 for b, and 70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-17±\sqrt{17^{2}-4\times 70}}{2}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 70}}{2}$$

Multiply -4 times 70.

$$x=\frac{-17±\sqrt{289-280}}{2}$$

Add 289 to -280.

$$x=\frac{-17±\sqrt{9}}{2}$$

Take the square root of 9.

$$x=\frac{-17±3}{2}$$

Now solve the equation x=\frac{-17±3}{2} when ± is plus. Add -17 to 3.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

Now solve the equation x=\frac{-17±3}{2} when ± is minus. Subtract 3 from -17.

$$x=-\frac{20}{2}$$

Divide -20 by 2.

$$x=-10$$

The equation is now solved.

$$x=-7 x=-10$$

Steps for Completing the Square

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

$$x^{2}+17x+70=0$$

Subtract 70 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+17x=-70$$

Divide 17, the coefficient of the x term, by 2 to get \frac{17}{2}. Then add the square of \frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+17x+\left(\frac{17}{2}\right)^{2}=-70+\left(\frac{17}{2}\right)^{2}$$

Square \frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+17x+\frac{289}{4}=-70+\frac{289}{4}$$

Add -70 to \frac{289}{4}.

$$x^{2}+17x+\frac{289}{4}=\frac{9}{4}$$

Factor x^{2}+17x+\frac{289}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{17}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x+\frac{17}{2}=\frac{3}{2} x+\frac{17}{2}=-\frac{3}{2}$$

Subtract \frac{17}{2} from both sides of the equation.

$$x=-7 x=-10$$