Solve 6a^2+51a=-63

$$6 a ^ { 2 } + 51 a = - 63$$

Solve for a

$$a=-7$$

$$a = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps Using Factoring By Grouping

Add 63 to both sides.

$$6a^{2}+51a+63=0$$

Divide both sides by 3.

$$2a^{2}+17a+21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2a^{2}+aa+ba+21. To find a and b, set up a system to be solved.

$$a+b=17 ab=2\times 21=42$$

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 42.

$$1,42 2,21 3,14 6,7$$

Calculate the sum for each pair.

$$1+42=43 2+21=23 3+14=17 6+7=13$$

The solution is the pair that gives sum 17.

$$a=3 b=14$$

Rewrite 2a^{2}+17a+21 as \left(2a^{2}+3a\right)+\left(14a+21\right).

$$\left(2a^{2}+3a\right)+\left(14a+21\right)$$

Factor out a in the first and 7 in the second group.

$$a\left(2a+3\right)+7\left(2a+3\right)$$

Factor out common term 2a+3 by using distributive property.

$$\left(2a+3\right)\left(a+7\right)$$

To find equation solutions, solve 2a+3=0 and a+7=0.

$$a=-\frac{3}{2} a=-7$$

Steps Using the Quadratic Formula

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.

$$6a^{2}+51a=-63$$

Add 63 to both sides of the equation.

$$6a^{2}+51a-\left(-63\right)=-63-\left(-63\right)$$

Subtracting -63 from itself leaves 0.

$$6a^{2}+51a-\left(-63\right)=0$$

Subtract -63 from 0.

$$6a^{2}+51a+63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 51 for b, and 63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-51±\sqrt{51^{2}-4\times 6\times 63}}{2\times 6}$$

Square 51.

$$a=\frac{-51±\sqrt{2601-4\times 6\times 63}}{2\times 6}$$

Multiply -4 times 6.

$$a=\frac{-51±\sqrt{2601-24\times 63}}{2\times 6}$$

Multiply -24 times 63.

$$a=\frac{-51±\sqrt{2601-1512}}{2\times 6}$$

Add 2601 to -1512.

$$a=\frac{-51±\sqrt{1089}}{2\times 6}$$

Take the square root of 1089.

$$a=\frac{-51±33}{2\times 6}$$

Multiply 2 times 6.

$$a=\frac{-51±33}{12}$$

Now solve the equation a=\frac{-51±33}{12} when ± is plus. Add -51 to 33.

$$a=-\frac{18}{12}$$

Reduce the fraction \frac{-18}{12} to lowest terms by extracting and canceling out 6.

$$a=-\frac{3}{2}$$

Now solve the equation a=\frac{-51±33}{12} when ± is minus. Subtract 33 from -51.

$$a=-\frac{84}{12}$$

Divide -84 by 12.

$$a=-7$$

The equation is now solved.

$$a=-\frac{3}{2} a=-7$$

Steps for Completing the Square

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.

$$6a^{2}+51a=-63$$

Divide both sides by 6.

$$\frac{6a^{2}+51a}{6}=-\frac{63}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$a^{2}+\frac{51}{6}a=-\frac{63}{6}$$

Reduce the fraction \frac{51}{6} to lowest terms by extracting and canceling out 3.

$$a^{2}+\frac{17}{2}a=-\frac{63}{6}$$

Reduce the fraction \frac{-63}{6} to lowest terms by extracting and canceling out 3.

$$a^{2}+\frac{17}{2}a=-\frac{21}{2}$$

Divide \frac{17}{2}, the coefficient of the x term, by 2 to get \frac{17}{4}. Then add the square of \frac{17}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{17}{2}a+\left(\frac{17}{4}\right)^{2}=-\frac{21}{2}+\left(\frac{17}{4}\right)^{2}$$

Square \frac{17}{4} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{17}{2}a+\frac{289}{16}=-\frac{21}{2}+\frac{289}{16}$$

Add -\frac{21}{2} to \frac{289}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{17}{2}a+\frac{289}{16}=\frac{121}{16}$$

Factor a^{2}+\frac{17}{2}a+\frac{289}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{17}{4}\right)^{2}=\frac{121}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{17}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Simplify.

$$a+\frac{17}{4}=\frac{11}{4} a+\frac{17}{4}=-\frac{11}{4}$$

Subtract \frac{17}{4} from both sides of the equation.

$$a=-\frac{3}{2} a=-7$$