Solve 30a^2=19a+63

$$30 a ^ { 2 } = 19 a + 63$$

Solve for a

$$a = -\frac{7}{6} = -1\frac{1}{6} \approx -1.166666667$$

$$a = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

Steps Using Factoring By Grouping

Subtract 19a from both sides.

$$30a^{2}-19a=63$$

Subtract 63 from both sides.

$$30a^{2}-19a-63=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 30a^{2}+aa+ba-63. To find a and b, set up a system to be solved.

$$a+b=-19 ab=30\left(-63\right)=-1890$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1890.

$$1,-1890 2,-945 3,-630 5,-378 6,-315 7,-270 9,-210 10,-189 14,-135 15,-126 18,-105 21,-90 27,-70 30,-63 35,-54 42,-45$$

Calculate the sum for each pair.

$$1-1890=-1889 2-945=-943 3-630=-627 5-378=-373 6-315=-309 7-270=-263 9-210=-201 10-189=-179 14-135=-121 15-126=-111 18-105=-87 21-90=-69 27-70=-43 30-63=-33 35-54=-19 42-45=-3$$

The solution is the pair that gives sum -19.

$$a=-54 b=35$$

Rewrite 30a^{2}-19a-63 as \left(30a^{2}-54a\right)+\left(35a-63\right).

$$\left(30a^{2}-54a\right)+\left(35a-63\right)$$

Factor out 6a in the first and 7 in the second group.

$$6a\left(5a-9\right)+7\left(5a-9\right)$$

Factor out common term 5a-9 by using distributive property.

$$\left(5a-9\right)\left(6a+7\right)$$

To find equation solutions, solve 5a-9=0 and 6a+7=0.

$$a=\frac{9}{5} a=-\frac{7}{6}$$

Steps Using the Quadratic Formula

Subtract 19a from both sides.

$$30a^{2}-19a=63$$

Subtract 63 from both sides.

$$30a^{2}-19a-63=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 30 for a, -19 for b, and -63 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 30\left(-63\right)}}{2\times 30}$$

Square -19.

$$a=\frac{-\left(-19\right)±\sqrt{361-4\times 30\left(-63\right)}}{2\times 30}$$

Multiply -4 times 30.

$$a=\frac{-\left(-19\right)±\sqrt{361-120\left(-63\right)}}{2\times 30}$$

Multiply -120 times -63.

$$a=\frac{-\left(-19\right)±\sqrt{361+7560}}{2\times 30}$$

Add 361 to 7560.

$$a=\frac{-\left(-19\right)±\sqrt{7921}}{2\times 30}$$

Take the square root of 7921.

$$a=\frac{-\left(-19\right)±89}{2\times 30}$$

The opposite of -19 is 19.

$$a=\frac{19±89}{2\times 30}$$

Multiply 2 times 30.

$$a=\frac{19±89}{60}$$

Now solve the equation a=\frac{19±89}{60} when ± is plus. Add 19 to 89.

$$a=\frac{108}{60}$$

Reduce the fraction \frac{108}{60} to lowest terms by extracting and canceling out 12.

$$a=\frac{9}{5}$$

Now solve the equation a=\frac{19±89}{60} when ± is minus. Subtract 89 from 19.

$$a=-\frac{70}{60}$$

Reduce the fraction \frac{-70}{60} to lowest terms by extracting and canceling out 10.

$$a=-\frac{7}{6}$$

The equation is now solved.

$$a=\frac{9}{5} a=-\frac{7}{6}$$

Steps for Completing the Square

Subtract 19a from both sides.

$$30a^{2}-19a=63$$

Divide both sides by 30.

$$\frac{30a^{2}-19a}{30}=\frac{63}{30}$$

Dividing by 30 undoes the multiplication by 30.

$$a^{2}-\frac{19}{30}a=\frac{63}{30}$$

Reduce the fraction \frac{63}{30} to lowest terms by extracting and canceling out 3.

$$a^{2}-\frac{19}{30}a=\frac{21}{10}$$

Divide -\frac{19}{30}, the coefficient of the x term, by 2 to get -\frac{19}{60}. Then add the square of -\frac{19}{60} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{19}{30}a+\left(-\frac{19}{60}\right)^{2}=\frac{21}{10}+\left(-\frac{19}{60}\right)^{2}$$

Square -\frac{19}{60} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{19}{30}a+\frac{361}{3600}=\frac{21}{10}+\frac{361}{3600}$$

Add \frac{21}{10} to \frac{361}{3600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{19}{30}a+\frac{361}{3600}=\frac{7921}{3600}$$

Factor a^{2}-\frac{19}{30}a+\frac{361}{3600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{19}{60}\right)^{2}=\frac{7921}{3600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{19}{60}\right)^{2}}=\sqrt{\frac{7921}{3600}}$$

Simplify.

$$a-\frac{19}{60}=\frac{89}{60} a-\frac{19}{60}=-\frac{89}{60}$$

Add \frac{19}{60} to both sides of the equation.

$$a=\frac{9}{5} a=-\frac{7}{6}$$