Solve 3x^2-23x+30=0

$$3 { x }^{ 2 } -23x+30=0$$

Solve for x

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

$$x=6$$

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 3x^{2}+ax+bx+30. To find a and b, set up a system to be solved.

$$a+b=-23 ab=3\times 30=90$$

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 -23.

$$a=-18 b=-5$$

Rewrite 3x^{2}-23x+30 as \left(3x^{2}-18x\right)+\left(-5x+30\right).

$$\left(3x^{2}-18x\right)+\left(-5x+30\right)$$

Factor out 3x in the first and -5 in the second group.

$$3x\left(x-6\right)-5\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(3x-5\right)$$

To find equation solutions, solve x-6=0 and 3x-5=0.

$$x=6 x=\frac{5}{3}$$

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.

$$3x^{2}-23x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -23 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 3\times 30}}{2\times 3}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\times 3\times 30}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-23\right)±\sqrt{529-12\times 30}}{2\times 3}$$

Multiply -12 times 30.

$$x=\frac{-\left(-23\right)±\sqrt{529-360}}{2\times 3}$$

Add 529 to -360.

$$x=\frac{-\left(-23\right)±\sqrt{169}}{2\times 3}$$

Take the square root of 169.

$$x=\frac{-\left(-23\right)±13}{2\times 3}$$

The opposite of -23 is 23.

$$x=\frac{23±13}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{23±13}{6}$$

Now solve the equation x=\frac{23±13}{6} when ± is plus. Add 23 to 13.

$$x=\frac{36}{6}$$

Divide 36 by 6.

$$x=6$$

Now solve the equation x=\frac{23±13}{6} when ± is minus. Subtract 13 from 23.

$$x=\frac{10}{6}$$

Reduce the fraction \frac{10}{6} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=6 x=\frac{5}{3}$$

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.

$$3x^{2}-23x+30=0$$

Subtract 30 from both sides of the equation.

$$3x^{2}-23x+30-30=-30$$

Subtracting 30 from itself leaves 0.

$$3x^{2}-23x=-30$$

Divide both sides by 3.

$$\frac{3x^{2}-23x}{3}=-\frac{30}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{23}{3}x=-\frac{30}{3}$$

Divide -30 by 3.

$$x^{2}-\frac{23}{3}x=-10$$

Divide -\frac{23}{3}, the coefficient of the x term, by 2 to get -\frac{23}{6}. Then add the square of -\frac{23}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{23}{3}x+\left(-\frac{23}{6}\right)^{2}=-10+\left(-\frac{23}{6}\right)^{2}$$

Square -\frac{23}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{23}{3}x+\frac{529}{36}=-10+\frac{529}{36}$$

Add -10 to \frac{529}{36}.

$$x^{2}-\frac{23}{3}x+\frac{529}{36}=\frac{169}{36}$$

Factor x^{2}-\frac{23}{3}x+\frac{529}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{23}{6}\right)^{2}=\frac{169}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{23}{6}\right)^{2}}=\sqrt{\frac{169}{36}}$$

Simplify.

$$x-\frac{23}{6}=\frac{13}{6} x-\frac{23}{6}=-\frac{13}{6}$$

Add \frac{23}{6} to both sides of the equation.

$$x=6 x=\frac{5}{3}$$