Solve 19x^2+15x-34=0

$$19 { x }^{ 2 } +15x-34=0$$

Solve for x

$$x = -\frac{34}{19} = -1\frac{15}{19} \approx -1.789473684$$

$$x=1$$

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 19x^{2}+ax+bx-34. To find a and b, set up a system to be solved.

$$a+b=15 ab=19\left(-34\right)=-646$$

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 -646.

$$-1,646 -2,323 -17,38 -19,34$$

Calculate the sum for each pair.

$$-1+646=645 -2+323=321 -17+38=21 -19+34=15$$

The solution is the pair that gives sum 15.

$$a=-19 b=34$$

Rewrite 19x^{2}+15x-34 as \left(19x^{2}-19x\right)+\left(34x-34\right).

$$\left(19x^{2}-19x\right)+\left(34x-34\right)$$

Factor out 19x in the first and 34 in the second group.

$$19x\left(x-1\right)+34\left(x-1\right)$$

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

$$\left(x-1\right)\left(19x+34\right)$$

To find equation solutions, solve x-1=0 and 19x+34=0.

$$x=1 x=-\frac{34}{19}$$

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.

$$19x^{2}+15x-34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 19 for a, 15 for b, and -34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 19\left(-34\right)}}{2\times 19}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 19\left(-34\right)}}{2\times 19}$$

Multiply -4 times 19.

$$x=\frac{-15±\sqrt{225-76\left(-34\right)}}{2\times 19}$$

Multiply -76 times -34.

$$x=\frac{-15±\sqrt{225+2584}}{2\times 19}$$

Add 225 to 2584.

$$x=\frac{-15±\sqrt{2809}}{2\times 19}$$

Take the square root of 2809.

$$x=\frac{-15±53}{2\times 19}$$

Multiply 2 times 19.

$$x=\frac{-15±53}{38}$$

Now solve the equation x=\frac{-15±53}{38} when ± is plus. Add -15 to 53.

$$x=\frac{38}{38}$$

Divide 38 by 38.

$$x=1$$

Now solve the equation x=\frac{-15±53}{38} when ± is minus. Subtract 53 from -15.

$$x=-\frac{68}{38}$$

Reduce the fraction \frac{-68}{38} to lowest terms by extracting and canceling out 2.

$$x=-\frac{34}{19}$$

The equation is now solved.

$$x=1 x=-\frac{34}{19}$$

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.

$$19x^{2}+15x-34=0$$

Add 34 to both sides of the equation.

$$19x^{2}+15x-34-\left(-34\right)=-\left(-34\right)$$

Subtracting -34 from itself leaves 0.

$$19x^{2}+15x=-\left(-34\right)$$

Subtract -34 from 0.

$$19x^{2}+15x=34$$

Divide both sides by 19.

$$\frac{19x^{2}+15x}{19}=\frac{34}{19}$$

Dividing by 19 undoes the multiplication by 19.

$$x^{2}+\frac{15}{19}x=\frac{34}{19}$$

Divide \frac{15}{19}, the coefficient of the x term, by 2 to get \frac{15}{38}. Then add the square of \frac{15}{38} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{15}{19}x+\left(\frac{15}{38}\right)^{2}=\frac{34}{19}+\left(\frac{15}{38}\right)^{2}$$

Square \frac{15}{38} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{15}{19}x+\frac{225}{1444}=\frac{34}{19}+\frac{225}{1444}$$

Add \frac{34}{19} to \frac{225}{1444} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{15}{19}x+\frac{225}{1444}=\frac{2809}{1444}$$

Factor x^{2}+\frac{15}{19}x+\frac{225}{1444}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{38}\right)^{2}=\frac{2809}{1444}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{38}\right)^{2}}=\sqrt{\frac{2809}{1444}}$$

Simplify.

$$x+\frac{15}{38}=\frac{53}{38} x+\frac{15}{38}=-\frac{53}{38}$$

Subtract \frac{15}{38} from both sides of the equation.

$$x=1 x=-\frac{34}{19}$$