Solve 6x^2-19x+15=0

$$6 { x }^{ 2 } -19x+15 = 0$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

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+15. To find a and b, set up a system to be solved.

$$a+b=-19 ab=6\times 15=90$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 90.

$$-1,-90 -2,-45 -3,-30 -5,-18 -6,-15 -9,-10$$

Calculate the sum for each pair.

$$-1-90=-91 -2-45=-47 -3-30=-33 -5-18=-23 -6-15=-21 -9-10=-19$$

The solution is the pair that gives sum -19.

$$a=-10 b=-9$$

Rewrite 6x^{2}-19x+15 as \left(6x^{2}-10x\right)+\left(-9x+15\right).

$$\left(6x^{2}-10x\right)+\left(-9x+15\right)$$

Factor out 2x in the first and -3 in the second group.

$$2x\left(3x-5\right)-3\left(3x-5\right)$$

Factor out common term 3x-5 by using distributive property.

$$\left(3x-5\right)\left(2x-3\right)$$

To find equation solutions, solve 3x-5=0 and 2x-3=0.

$$x=\frac{5}{3} x=\frac{3}{2}$$

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.

$$6x^{2}-19x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -19 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 6\times 15}}{2\times 6}$$

Square -19.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 6\times 15}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-19\right)±\sqrt{361-24\times 15}}{2\times 6}$$

Multiply -24 times 15.

$$x=\frac{-\left(-19\right)±\sqrt{361-360}}{2\times 6}$$

Add 361 to -360.

$$x=\frac{-\left(-19\right)±\sqrt{1}}{2\times 6}$$

Take the square root of 1.

$$x=\frac{-\left(-19\right)±1}{2\times 6}$$

The opposite of -19 is 19.

$$x=\frac{19±1}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{19±1}{12}$$

Now solve the equation x=\frac{19±1}{12} when ± is plus. Add 19 to 1.

$$x=\frac{20}{12}$$

Reduce the fraction \frac{20}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{5}{3}$$

Now solve the equation x=\frac{19±1}{12} when ± is minus. Subtract 1 from 19.

$$x=\frac{18}{12}$$

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

$$x=\frac{3}{2}$$

The equation is now solved.

$$x=\frac{5}{3} x=\frac{3}{2}$$

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.

$$6x^{2}-19x+15=0$$

Subtract 15 from both sides of the equation.

$$6x^{2}-19x+15-15=-15$$

Subtracting 15 from itself leaves 0.

$$6x^{2}-19x=-15$$

Divide both sides by 6.

$$\frac{6x^{2}-19x}{6}=-\frac{15}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{19}{6}x=-\frac{15}{6}$$

Reduce the fraction \frac{-15}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{19}{6}x=-\frac{5}{2}$$

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.

$$x^{2}-\frac{19}{6}x+\left(-\frac{19}{12}\right)^{2}=-\frac{5}{2}+\left(-\frac{19}{12}\right)^{2}$$

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

$$x^{2}-\frac{19}{6}x+\frac{361}{144}=-\frac{5}{2}+\frac{361}{144}$$

Add -\frac{5}{2} to \frac{361}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{19}{6}x+\frac{361}{144}=\frac{1}{144}$$

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}.

$$\left(x-\frac{19}{12}\right)^{2}=\frac{1}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{12}\right)^{2}}=\sqrt{\frac{1}{144}}$$

Simplify.

$$x-\frac{19}{12}=\frac{1}{12} x-\frac{19}{12}=-\frac{1}{12}$$

Add \frac{19}{12} to both sides of the equation.

$$x=\frac{5}{3} x=\frac{3}{2}$$