Solve 6x^2+19x-7=0

$$6 { x }^{ 2 } +19x-7=0$$

Solve for x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$x=\frac{1}{3}\approx 0.333333333$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 6x^{2}+ax+bx-7. To find a and b, set up a system to be solved.

$$a+b=19 ab=6\left(-7\right)=-42$$

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 -42.

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

Calculate the sum for each pair.

$$-1+42=41 -2+21=19 -3+14=11 -6+7=1$$

The solution is the pair that gives sum 19.

$$a=-2 b=21$$

Rewrite 6x^{2}+19x-7 as \left(6x^{2}-2x\right)+\left(21x-7\right).

$$\left(6x^{2}-2x\right)+\left(21x-7\right)$$

Factor out 2x in the first and 7 in the second group.

$$2x\left(3x-1\right)+7\left(3x-1\right)$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(2x+7\right)$$

To find equation solutions, solve 3x-1=0 and 2x+7=0.

$$x=\frac{1}{3} x=-\frac{7}{2}$$

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.

$$6x^{2}+19x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 19 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\times 6\left(-7\right)}}{2\times 6}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\times 6\left(-7\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-19±\sqrt{361-24\left(-7\right)}}{2\times 6}$$

Multiply -24 times -7.

$$x=\frac{-19±\sqrt{361+168}}{2\times 6}$$

Add 361 to 168.

$$x=\frac{-19±\sqrt{529}}{2\times 6}$$

Take the square root of 529.

$$x=\frac{-19±23}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-19±23}{12}$$

Now solve the equation x=\frac{-19±23}{12} when ± is plus. Add -19 to 23.

$$x=\frac{4}{12}$$

Reduce the fraction \frac{4}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{-19±23}{12} when ± is minus. Subtract 23 from -19.

$$x=-\frac{42}{12}$$

Reduce the fraction \frac{-42}{12} to lowest terms by extracting and canceling out 6.

$$x=-\frac{7}{2}$$

The equation is now solved.

$$x=\frac{1}{3} x=-\frac{7}{2}$$

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.

$$6x^{2}+19x-7=0$$

Add 7 to both sides of the equation.

$$6x^{2}+19x-7-\left(-7\right)=-\left(-7\right)$$

Subtracting -7 from itself leaves 0.

$$6x^{2}+19x=-\left(-7\right)$$

Subtract -7 from 0.

$$6x^{2}+19x=7$$

Divide both sides by 6.

$$\frac{6x^{2}+19x}{6}=\frac{7}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{19}{6}x=\frac{7}{6}$$

Divide \frac{19}{6}, the coefficient of the x term, by 2 to get \frac{19}{12}. Then add the square of \frac{19}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{19}{6}x+\left(\frac{19}{12}\right)^{2}=\frac{7}{6}+\left(\frac{19}{12}\right)^{2}$$

Square \frac{19}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{19}{6}x+\frac{361}{144}=\frac{7}{6}+\frac{361}{144}$$

Add \frac{7}{6} to \frac{361}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{19}{6}x+\frac{361}{144}=\frac{529}{144}$$

Factor x^{2}+\frac{19}{6}x+\frac{361}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{19}{12}\right)^{2}=\frac{529}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{19}{12}\right)^{2}}=\sqrt{\frac{529}{144}}$$

Simplify.

$$x+\frac{19}{12}=\frac{23}{12} x+\frac{19}{12}=-\frac{23}{12}$$

Subtract \frac{19}{12} from both sides of the equation.

$$x=\frac{1}{3} x=-\frac{7}{2}$$