Solve v^2+7v-60

$$v ^ { 2 } + 7 v - 60$$

Factor

$$\left(v-5\right)\left(v+12\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as v^{2}+av+bv-60. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\left(-60\right)=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 7.

$$a=-5 b=12$$

Rewrite v^{2}+7v-60 as \left(v^{2}-5v\right)+\left(12v-60\right).

$$\left(v^{2}-5v\right)+\left(12v-60\right)$$

Factor out v in the first and 12 in the second group.

$$v\left(v-5\right)+12\left(v-5\right)$$

Factor out common term v-5 by using distributive property.

$$\left(v-5\right)\left(v+12\right)$$

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.

$$v^{2}+7v-60=0$$

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.

$$v=\frac{-7±\sqrt{7^{2}-4\left(-60\right)}}{2}$$

Square 7.

$$v=\frac{-7±\sqrt{49-4\left(-60\right)}}{2}$$

Multiply -4 times -60.

$$v=\frac{-7±\sqrt{49+240}}{2}$$

Add 49 to 240.

$$v=\frac{-7±\sqrt{289}}{2}$$

Take the square root of 289.

$$v=\frac{-7±17}{2}$$

Now solve the equation v=\frac{-7±17}{2} when ± is plus. Add -7 to 17.

$$v=\frac{10}{2}$$

Divide 10 by 2.

$$v=5$$

Now solve the equation v=\frac{-7±17}{2} when ± is minus. Subtract 17 from -7.

$$v=-\frac{24}{2}$$

Divide -24 by 2.

$$v=-12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and -12 for x_{2}.

$$v^{2}+7v-60=\left(v-5\right)\left(v-\left(-12\right)\right)$$

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

$$v^{2}+7v-60=\left(v-5\right)\left(v+12\right)$$

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.

$$x ^ 2 +7x -60 = 0$$

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

$$r + s = -7 rs = -60$$

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 = -60

$$(-\frac{7}{2} - u) (-\frac{7}{2} + u) = -60$$

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

$$\frac{49}{4} - u^2 = -60$$

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

$$-u^2 = -60-\frac{49}{4} = -\frac{289}{4}$$

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

$$u^2 = \frac{289}{4} u = \pm\sqrt{\frac{289}{4}} = \pm \frac{17}{2}$$

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

$$r =-\frac{7}{2} - \frac{17}{2} = -12 s = -\frac{7}{2} + \frac{17}{2} = 5$$

Evaluate

$$\left(v-5\right)\left(v+12\right)$$