Solve 7k^2+15k+8

$$7 k ^ { 2 } + 15 k + 8$$

Factor

$$\left(k+1\right)\left(7k+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7k^{2}+ak+bk+8. To find a and b, set up a system to be solved.

$$a+b=15 ab=7\times 8=56$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 56.

$$1,56 2,28 4,14 7,8$$

Calculate the sum for each pair.

$$1+56=57 2+28=30 4+14=18 7+8=15$$

The solution is the pair that gives sum 15.

$$a=7 b=8$$

Rewrite 7k^{2}+15k+8 as \left(7k^{2}+7k\right)+\left(8k+8\right).

$$\left(7k^{2}+7k\right)+\left(8k+8\right)$$

Factor out 7k in the first and 8 in the second group.

$$7k\left(k+1\right)+8\left(k+1\right)$$

Factor out common term k+1 by using distributive property.

$$\left(k+1\right)\left(7k+8\right)$$

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.

$$7k^{2}+15k+8=0$$

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.

$$k=\frac{-15±\sqrt{15^{2}-4\times 7\times 8}}{2\times 7}$$

Square 15.

$$k=\frac{-15±\sqrt{225-4\times 7\times 8}}{2\times 7}$$

Multiply -4 times 7.

$$k=\frac{-15±\sqrt{225-28\times 8}}{2\times 7}$$

Multiply -28 times 8.

$$k=\frac{-15±\sqrt{225-224}}{2\times 7}$$

Add 225 to -224.

$$k=\frac{-15±\sqrt{1}}{2\times 7}$$

Take the square root of 1.

$$k=\frac{-15±1}{2\times 7}$$

Multiply 2 times 7.

$$k=\frac{-15±1}{14}$$

Now solve the equation k=\frac{-15±1}{14} when ± is plus. Add -15 to 1.

$$k=-\frac{14}{14}$$

Divide -14 by 14.

$$k=-1$$

Now solve the equation k=\frac{-15±1}{14} when ± is minus. Subtract 1 from -15.

$$k=-\frac{16}{14}$$

Reduce the fraction \frac{-16}{14} to lowest terms by extracting and canceling out 2.

$$k=-\frac{8}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -\frac{8}{7} for x_{2}.

$$7k^{2}+15k+8=7\left(k-\left(-1\right)\right)\left(k-\left(-\frac{8}{7}\right)\right)$$

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

$$7k^{2}+15k+8=7\left(k+1\right)\left(k+\frac{8}{7}\right)$$

Add \frac{8}{7} to k by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7k^{2}+15k+8=7\left(k+1\right)\times \frac{7k+8}{7}$$

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

$$7k^{2}+15k+8=\left(k+1\right)\left(7k+8\right)$$

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

$$x ^ 2 +\frac{15}{7}x +\frac{8}{7} = 0$$

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

$$r + s = -\frac{15}{7} rs = \frac{8}{7}$$

Two numbers r and s sum up to -\frac{15}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{15}{7} = -\frac{15}{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>

$$r = -\frac{15}{14} - u s = -\frac{15}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{7}

$$(-\frac{15}{14} - u) (-\frac{15}{14} + u) = \frac{8}{7}$$

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

$$\frac{225}{196} - u^2 = \frac{8}{7}$$

Simplify the expression by subtracting \frac{225}{196} on both sides

$$-u^2 = \frac{8}{7}-\frac{225}{196} = -\frac{1}{196}$$

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

$$u^2 = \frac{1}{196} u = \pm\sqrt{\frac{1}{196}} = \pm \frac{1}{14}$$

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

$$r =-\frac{15}{14} - \frac{1}{14} = -1.143 s = -\frac{15}{14} + \frac{1}{14} = -1.000$$

Evaluate

$$\left(k+1\right)\left(7k+8\right)$$