Solve 2a^2-9a-45

$$2 a ^ { 2 } - 9 a - 45$$

Factor

$$\left(2a-15\right)\left(a+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2a^{2}+pa+qa-45. To find p and q, set up a system to be solved.

$$p+q=-9 pq=2\left(-45\right)=-90$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. 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.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

The solution is the pair that gives sum -9.

$$p=-15 q=6$$

Rewrite 2a^{2}-9a-45 as \left(2a^{2}-15a\right)+\left(6a-45\right).

$$\left(2a^{2}-15a\right)+\left(6a-45\right)$$

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

$$a\left(2a-15\right)+3\left(2a-15\right)$$

Factor out common term 2a-15 by using distributive property.

$$\left(2a-15\right)\left(a+3\right)$$

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.

$$2a^{2}-9a-45=0$$

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.

$$a=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\left(-45\right)}}{2\times 2}$$

Square -9.

$$a=\frac{-\left(-9\right)±\sqrt{81-4\times 2\left(-45\right)}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-\left(-9\right)±\sqrt{81-8\left(-45\right)}}{2\times 2}$$

Multiply -8 times -45.

$$a=\frac{-\left(-9\right)±\sqrt{81+360}}{2\times 2}$$

Add 81 to 360.

$$a=\frac{-\left(-9\right)±\sqrt{441}}{2\times 2}$$

Take the square root of 441.

$$a=\frac{-\left(-9\right)±21}{2\times 2}$$

The opposite of -9 is 9.

$$a=\frac{9±21}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{9±21}{4}$$

Now solve the equation a=\frac{9±21}{4} when ± is plus. Add 9 to 21.

$$a=\frac{30}{4}$$

Reduce the fraction \frac{30}{4} to lowest terms by extracting and canceling out 2.

$$a=\frac{15}{2}$$

Now solve the equation a=\frac{9±21}{4} when ± is minus. Subtract 21 from 9.

$$a=-\frac{12}{4}$$

Divide -12 by 4.

$$a=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{15}{2} for x_{1} and -3 for x_{2}.

$$2a^{2}-9a-45=2\left(a-\frac{15}{2}\right)\left(a-\left(-3\right)\right)$$

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

$$2a^{2}-9a-45=2\left(a-\frac{15}{2}\right)\left(a+3\right)$$

Subtract \frac{15}{2} from a by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$2a^{2}-9a-45=2\times \frac{2a-15}{2}\left(a+3\right)$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2a^{2}-9a-45=\left(2a-15\right)\left(a+3\right)$$

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 2

$$x ^ 2 -\frac{9}{2}x -\frac{45}{2} = 0$$

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

$$r + s = \frac{9}{2} rs = -\frac{45}{2}$$

Two numbers r and s sum up to \frac{9}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{9}{2} = \frac{9}{4}. 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>

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

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{45}{2}

$$(\frac{9}{4} - u) (\frac{9}{4} + u) = -\frac{45}{2}$$

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

$$\frac{81}{16} - u^2 = -\frac{45}{2}$$

Simplify the expression by subtracting \frac{81}{16} on both sides

$$-u^2 = -\frac{45}{2}-\frac{81}{16} = -\frac{441}{16}$$

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

$$u^2 = \frac{441}{16} u = \pm\sqrt{\frac{441}{16}} = \pm \frac{21}{4}$$

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

$$r =\frac{9}{4} - \frac{21}{4} = -3 s = \frac{9}{4} + \frac{21}{4} = 7.500$$

Evaluate

$$\left(2a-15\right)\left(a+3\right)$$