Solve 5v^2+45v+70

$$5 v ^ { 2 } + 45 v + 70$$

Factor

$$5\left(v+2\right)\left(v+7\right)$$

Solution Steps

Factor out 5.

$$5\left(v^{2}+9v+14\right)$$

Consider v^{2}+9v+14. Factor the expression by grouping. First, the expression needs to be rewritten as v^{2}+av+bv+14. To find a and b, set up a system to be solved.

$$a+b=9 ab=1\times 14=14$$

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 14.

$$1,14 2,7$$

Calculate the sum for each pair.

$$1+14=15 2+7=9$$

The solution is the pair that gives sum 9.

$$a=2 b=7$$

Rewrite v^{2}+9v+14 as \left(v^{2}+2v\right)+\left(7v+14\right).

$$\left(v^{2}+2v\right)+\left(7v+14\right)$$

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

$$v\left(v+2\right)+7\left(v+2\right)$$

Factor out common term v+2 by using distributive property.

$$\left(v+2\right)\left(v+7\right)$$

Rewrite the complete factored expression.

$$5\left(v+2\right)\left(v+7\right)$$

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.

$$5v^{2}+45v+70=0$$

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.

$$v=\frac{-45±\sqrt{45^{2}-4\times 5\times 70}}{2\times 5}$$

Square 45.

$$v=\frac{-45±\sqrt{2025-4\times 5\times 70}}{2\times 5}$$

Multiply -4 times 5.

$$v=\frac{-45±\sqrt{2025-20\times 70}}{2\times 5}$$

Multiply -20 times 70.

$$v=\frac{-45±\sqrt{2025-1400}}{2\times 5}$$

Add 2025 to -1400.

$$v=\frac{-45±\sqrt{625}}{2\times 5}$$

Take the square root of 625.

$$v=\frac{-45±25}{2\times 5}$$

Multiply 2 times 5.

$$v=\frac{-45±25}{10}$$

Now solve the equation v=\frac{-45±25}{10} when ± is plus. Add -45 to 25.

$$v=-\frac{20}{10}$$

Divide -20 by 10.

$$v=-2$$

Now solve the equation v=\frac{-45±25}{10} when ± is minus. Subtract 25 from -45.

$$v=-\frac{70}{10}$$

Divide -70 by 10.

$$v=-7$$

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

$$5v^{2}+45v+70=5\left(v-\left(-2\right)\right)\left(v-\left(-7\right)\right)$$

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

$$5v^{2}+45v+70=5\left(v+2\right)\left(v+7\right)$$

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 5

$$x ^ 2 +9x +14 = 0$$

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

$$r + s = -9 rs = 14$$

Two numbers r and s sum up to -9 exactly when the average of the two numbers is \frac{1}{2}*-9 = -\frac{9}{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>

$$r = -\frac{9}{2} - u s = -\frac{9}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 14

$$(-\frac{9}{2} - u) (-\frac{9}{2} + u) = 14$$

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

$$\frac{81}{4} - u^2 = 14$$

Simplify the expression by subtracting \frac{81}{4} on both sides

$$-u^2 = 14-\frac{81}{4} = -\frac{25}{4}$$

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

$$u^2 = \frac{25}{4} u = \pm\sqrt{\frac{25}{4}} = \pm \frac{5}{2}$$

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

$$r =-\frac{9}{2} - \frac{5}{2} = -7 s = -\frac{9}{2} + \frac{5}{2} = -2$$

Evaluate

$$5\left(v+2\right)\left(v+7\right)$$