Solve 7a^2+47a-14

$$7 a ^ { 2 } + 47 a - 14$$

Factor

$$\left(7a-2\right)\left(a+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7a^{2}+pa+qa-14. To find p and q, set up a system to be solved.

$$p+q=47 pq=7\left(-14\right)=-98$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -98.

$$-1,98 -2,49 -7,14$$

Calculate the sum for each pair.

$$-1+98=97 -2+49=47 -7+14=7$$

The solution is the pair that gives sum 47.

$$p=-2 q=49$$

Rewrite 7a^{2}+47a-14 as \left(7a^{2}-2a\right)+\left(49a-14\right).

$$\left(7a^{2}-2a\right)+\left(49a-14\right)$$

Factor out a in the first and 7 in the second group.

$$a\left(7a-2\right)+7\left(7a-2\right)$$

Factor out common term 7a-2 by using distributive property.

$$\left(7a-2\right)\left(a+7\right)$$

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.

$$7a^{2}+47a-14=0$$

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.

$$a=\frac{-47±\sqrt{47^{2}-4\times 7\left(-14\right)}}{2\times 7}$$

Square 47.

$$a=\frac{-47±\sqrt{2209-4\times 7\left(-14\right)}}{2\times 7}$$

Multiply -4 times 7.

$$a=\frac{-47±\sqrt{2209-28\left(-14\right)}}{2\times 7}$$

Multiply -28 times -14.

$$a=\frac{-47±\sqrt{2209+392}}{2\times 7}$$

Add 2209 to 392.

$$a=\frac{-47±\sqrt{2601}}{2\times 7}$$

Take the square root of 2601.

$$a=\frac{-47±51}{2\times 7}$$

Multiply 2 times 7.

$$a=\frac{-47±51}{14}$$

Now solve the equation a=\frac{-47±51}{14} when ± is plus. Add -47 to 51.

$$a=\frac{4}{14}$$

Reduce the fraction \frac{4}{14} to lowest terms by extracting and canceling out 2.

$$a=\frac{2}{7}$$

Now solve the equation a=\frac{-47±51}{14} when ± is minus. Subtract 51 from -47.

$$a=-\frac{98}{14}$$

Divide -98 by 14.

$$a=-7$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{7} for x_{1} and -7 for x_{2}.

$$7a^{2}+47a-14=7\left(a-\frac{2}{7}\right)\left(a-\left(-7\right)\right)$$

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

$$7a^{2}+47a-14=7\left(a-\frac{2}{7}\right)\left(a+7\right)$$

Subtract \frac{2}{7} from a by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$7a^{2}+47a-14=7\times \frac{7a-2}{7}\left(a+7\right)$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7a^{2}+47a-14=\left(7a-2\right)\left(a+7\right)$$

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 7

$$x ^ 2 +\frac{47}{7}x -2 = 0$$

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

$$r + s = -\frac{47}{7} rs = -2$$

Two numbers r and s sum up to -\frac{47}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{47}{7} = -\frac{47}{14}. 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>

$$r = -\frac{47}{14} - u s = -\frac{47}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -2

$$(-\frac{47}{14} - u) (-\frac{47}{14} + u) = -2$$

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

$$\frac{2209}{196} - u^2 = -2$$

Simplify the expression by subtracting \frac{2209}{196} on both sides

$$-u^2 = -2-\frac{2209}{196} = -\frac{2601}{196}$$

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

$$u^2 = \frac{2601}{196} u = \pm\sqrt{\frac{2601}{196}} = \pm \frac{51}{14}$$

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

$$r =-\frac{47}{14} - \frac{51}{14} = -7 s = -\frac{47}{14} + \frac{51}{14} = 0.286$$

Evaluate

$$\left(7a-2\right)\left(a+7\right)$$