Solve 4y^2-36y-40

$$4 y ^ { 2 } - 36 y - 40$$

Factor

$$4\left(y-10\right)\left(y+1\right)$$

Solution Steps

Factor out 4.

$$4\left(y^{2}-9y-10\right)$$

Consider y^{2}-9y-10. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-10. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\left(-10\right)=-10$$

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 -10.

$$1,-10 2,-5$$

Calculate the sum for each pair.

$$1-10=-9 2-5=-3$$

The solution is the pair that gives sum -9.

$$a=-10 b=1$$

Rewrite y^{2}-9y-10 as \left(y^{2}-10y\right)+\left(y-10\right).

$$\left(y^{2}-10y\right)+\left(y-10\right)$$

Factor out y in y^{2}-10y.

$$y\left(y-10\right)+y-10$$

Factor out common term y-10 by using distributive property.

$$\left(y-10\right)\left(y+1\right)$$

Rewrite the complete factored expression.

$$4\left(y-10\right)\left(y+1\right)$$

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.

$$4y^{2}-36y-40=0$$

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.

$$y=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 4\left(-40\right)}}{2\times 4}$$

Square -36.

$$y=\frac{-\left(-36\right)±\sqrt{1296-4\times 4\left(-40\right)}}{2\times 4}$$

Multiply -4 times 4.

$$y=\frac{-\left(-36\right)±\sqrt{1296-16\left(-40\right)}}{2\times 4}$$

Multiply -16 times -40.

$$y=\frac{-\left(-36\right)±\sqrt{1296+640}}{2\times 4}$$

Add 1296 to 640.

$$y=\frac{-\left(-36\right)±\sqrt{1936}}{2\times 4}$$

Take the square root of 1936.

$$y=\frac{-\left(-36\right)±44}{2\times 4}$$

The opposite of -36 is 36.

$$y=\frac{36±44}{2\times 4}$$

Multiply 2 times 4.

$$y=\frac{36±44}{8}$$

Now solve the equation y=\frac{36±44}{8} when ± is plus. Add 36 to 44.

$$y=\frac{80}{8}$$

Divide 80 by 8.

$$y=10$$

Now solve the equation y=\frac{36±44}{8} when ± is minus. Subtract 44 from 36.

$$y=-\frac{8}{8}$$

Divide -8 by 8.

$$y=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 10 for x_{1} and -1 for x_{2}.

$$4y^{2}-36y-40=4\left(y-10\right)\left(y-\left(-1\right)\right)$$

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

$$4y^{2}-36y-40=4\left(y-10\right)\left(y+1\right)$$

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 4

$$x ^ 2 -9x -10 = 0$$

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

$$r + s = 9 rs = -10$$

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-gzdabgg4ehffg0hf.b01.azurefd.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 = -10

$$(\frac{9}{2} - u) (\frac{9}{2} + u) = -10$$

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

$$\frac{81}{4} - u^2 = -10$$

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

$$-u^2 = -10-\frac{81}{4} = -\frac{121}{4}$$

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

$$u^2 = \frac{121}{4} u = \pm\sqrt{\frac{121}{4}} = \pm \frac{11}{2}$$

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

$$r =\frac{9}{2} - \frac{11}{2} = -1 s = \frac{9}{2} + \frac{11}{2} = 10$$

Evaluate

$$4\left(y-10\right)\left(y+1\right)$$