Solve 4m^2+9m-9

$$4 m ^ { 2 } + 9 m - 9$$

Factor

$$\left(4m-3\right)\left(m+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4m^{2}+am+bm-9. To find a and b, set up a system to be solved.

$$a+b=9 ab=4\left(-9\right)=-36$$

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 -36.

$$-1,36 -2,18 -3,12 -4,9 -6,6$$

Calculate the sum for each pair.

$$-1+36=35 -2+18=16 -3+12=9 -4+9=5 -6+6=0$$

The solution is the pair that gives sum 9.

$$a=-3 b=12$$

Rewrite 4m^{2}+9m-9 as \left(4m^{2}-3m\right)+\left(12m-9\right).

$$\left(4m^{2}-3m\right)+\left(12m-9\right)$$

Factor out m in the first and 3 in the second group.

$$m\left(4m-3\right)+3\left(4m-3\right)$$

Factor out common term 4m-3 by using distributive property.

$$\left(4m-3\right)\left(m+3\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$4m^{2}+9m-9=0$$

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.

$$m=\frac{-9±\sqrt{9^{2}-4\times 4\left(-9\right)}}{2\times 4}$$

Square 9.

$$m=\frac{-9±\sqrt{81-4\times 4\left(-9\right)}}{2\times 4}$$

Multiply -4 times 4.

$$m=\frac{-9±\sqrt{81-16\left(-9\right)}}{2\times 4}$$

Multiply -16 times -9.

$$m=\frac{-9±\sqrt{81+144}}{2\times 4}$$

Add 81 to 144.

$$m=\frac{-9±\sqrt{225}}{2\times 4}$$

Take the square root of 225.

$$m=\frac{-9±15}{2\times 4}$$

Multiply 2 times 4.

$$m=\frac{-9±15}{8}$$

Now solve the equation m=\frac{-9±15}{8} when ± is plus. Add -9 to 15.

$$m=\frac{6}{8}$$

Reduce the fraction \frac{6}{8} to lowest terms by extracting and canceling out 2.

$$m=\frac{3}{4}$$

Now solve the equation m=\frac{-9±15}{8} when ± is minus. Subtract 15 from -9.

$$m=-\frac{24}{8}$$

Divide -24 by 8.

$$m=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{4} for x_{1} and -3 for x_{2}.

$$4m^{2}+9m-9=4\left(m-\frac{3}{4}\right)\left(m-\left(-3\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$4m^{2}+9m-9=4\left(m-\frac{3}{4}\right)\left(m+3\right)$$

Subtract \frac{3}{4} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4m^{2}+9m-9=4\times \frac{4m-3}{4}\left(m+3\right)$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4m^{2}+9m-9=\left(4m-3\right)\left(m+3\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 4

$$x ^ 2 +\frac{9}{4}x -\frac{9}{4} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -\frac{9}{4} rs = -\frac{9}{4}$$

Two numbers r and s sum up to -\frac{9}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{9}{4} = -\frac{9}{8}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -\frac{9}{8} - u s = -\frac{9}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{4}

$$(-\frac{9}{8} - u) (-\frac{9}{8} + u) = -\frac{9}{4}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{81}{64} - u^2 = -\frac{9}{4}$$

Simplify the expression by subtracting \frac{81}{64} on both sides

$$-u^2 = -\frac{9}{4}-\frac{81}{64} = -\frac{225}{64}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{225}{64} u = \pm\sqrt{\frac{225}{64}} = \pm \frac{15}{8}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-\frac{9}{8} - \frac{15}{8} = -3 s = -\frac{9}{8} + \frac{15}{8} = 0.750$$

Evaluate

$$\left(4m-3\right)\left(m+3\right)$$