Solve 2a^2-14a+24

$$2 a ^ { 2 } - 14 a + 24$$

Factor

$$2\left(a-4\right)\left(a-3\right)$$

Solution Steps

Factor out 2.

$$2\left(a^{2}-7a+12\right)$$

Consider a^{2}-7a+12. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa+12. To find p and q, set up a system to be solved.

$$p+q=-7 pq=1\times 12=12$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -7.

$$p=-4 q=-3$$

Rewrite a^{2}-7a+12 as \left(a^{2}-4a\right)+\left(-3a+12\right).

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

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

$$a\left(a-4\right)-3\left(a-4\right)$$

Factor out common term a-4 by using distributive property.

$$\left(a-4\right)\left(a-3\right)$$

Rewrite the complete factored expression.

$$2\left(a-4\right)\left(a-3\right)$$

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.

$$2a^{2}-14a+24=0$$

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.

$$a=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 2\times 24}}{2\times 2}$$

Square -14.

$$a=\frac{-\left(-14\right)±\sqrt{196-4\times 2\times 24}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-\left(-14\right)±\sqrt{196-8\times 24}}{2\times 2}$$

Multiply -8 times 24.

$$a=\frac{-\left(-14\right)±\sqrt{196-192}}{2\times 2}$$

Add 196 to -192.

$$a=\frac{-\left(-14\right)±\sqrt{4}}{2\times 2}$$

Take the square root of 4.

$$a=\frac{-\left(-14\right)±2}{2\times 2}$$

The opposite of -14 is 14.

$$a=\frac{14±2}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{14±2}{4}$$

Now solve the equation a=\frac{14±2}{4} when ± is plus. Add 14 to 2.

$$a=\frac{16}{4}$$

Divide 16 by 4.

$$a=4$$

Now solve the equation a=\frac{14±2}{4} when ± is minus. Subtract 2 from 14.

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

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 4 for x_{1} and 3 for x_{2}.

$$2a^{2}-14a+24=2\left(a-4\right)\left(a-3\right)$$

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

$$x ^ 2 -7x +12 = 0$$

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

$$r + s = 7 rs = 12$$

Two numbers r and s sum up to 7 exactly when the average of the two numbers is \frac{1}{2}*7 = \frac{7}{2}. 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>

$$r = \frac{7}{2} - u s = \frac{7}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 12

$$(\frac{7}{2} - u) (\frac{7}{2} + u) = 12$$

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

$$\frac{49}{4} - u^2 = 12$$

Simplify the expression by subtracting \frac{49}{4} on both sides

$$-u^2 = 12-\frac{49}{4} = -\frac{1}{4}$$

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

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

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

$$r =\frac{7}{2} - \frac{1}{2} = 3 s = \frac{7}{2} + \frac{1}{2} = 4$$

Evaluate

$$2\left(a-4\right)\left(a-3\right)$$