Solve y^2-14y-51

$$y ^ { 2 } - 14 y - 51$$

Factor

$$\left(y-17\right)\left(y+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-51. To find a and b, set up a system to be solved.

$$a+b=-14 ab=1\left(-51\right)=-51$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -51.

$$1,-51 3,-17$$

Calculate the sum for each pair.

$$1-51=-50 3-17=-14$$

The solution is the pair that gives sum -14.

$$a=-17 b=3$$

Rewrite y^{2}-14y-51 as \left(y^{2}-17y\right)+\left(3y-51\right).

$$\left(y^{2}-17y\right)+\left(3y-51\right)$$

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

$$y\left(y-17\right)+3\left(y-17\right)$$

Factor out common term y-17 by using distributive property.

$$\left(y-17\right)\left(y+3\right)$$

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.

$$y^{2}-14y-51=0$$

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.

$$y=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-51\right)}}{2}$$

Square -14.

$$y=\frac{-\left(-14\right)±\sqrt{196-4\left(-51\right)}}{2}$$

Multiply -4 times -51.

$$y=\frac{-\left(-14\right)±\sqrt{196+204}}{2}$$

Add 196 to 204.

$$y=\frac{-\left(-14\right)±\sqrt{400}}{2}$$

Take the square root of 400.

$$y=\frac{-\left(-14\right)±20}{2}$$

The opposite of -14 is 14.

$$y=\frac{14±20}{2}$$

Now solve the equation y=\frac{14±20}{2} when ± is plus. Add 14 to 20.

$$y=\frac{34}{2}$$

Divide 34 by 2.

$$y=17$$

Now solve the equation y=\frac{14±20}{2} when ± is minus. Subtract 20 from 14.

$$y=-\frac{6}{2}$$

Divide -6 by 2.

$$y=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 17 for x_{1} and -3 for x_{2}.

$$y^{2}-14y-51=\left(y-17\right)\left(y-\left(-3\right)\right)$$

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

$$y^{2}-14y-51=\left(y-17\right)\left(y+3\right)$$

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.

$$x ^ 2 -14x -51 = 0$$

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

$$r + s = 14 rs = -51$$

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

$$r = 7 - u s = 7 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -51

$$(7 - u) (7 + u) = -51$$

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

$$49 - u^2 = -51$$

Simplify the expression by subtracting 49 on both sides

$$-u^2 = -51-49 = -100$$

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

$$u^2 = 100 u = \pm\sqrt{100} = \pm 10$$

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

$$r =7 - 10 = -3 s = 7 + 10 = 17$$

Evaluate

$$\left(y-17\right)\left(y+3\right)$$