Solve 7y^2+77y+70=

$$7 y ^ { 2 } + 77 y + 70 =$$

Factor

$$7\left(y+1\right)\left(y+10\right)$$

Solution Steps

Factor out 7.

$$7\left(y^{2}+11y+10\right)$$

Consider y^{2}+11y+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.

$$a+b=11 ab=1\times 10=10$$

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 10.

$$1,10 2,5$$

Calculate the sum for each pair.

$$1+10=11 2+5=7$$

The solution is the pair that gives sum 11.

$$a=1 b=10$$

Rewrite y^{2}+11y+10 as \left(y^{2}+y\right)+\left(10y+10\right).

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

Factor out y in the first and 10 in the second group.

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

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

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

Rewrite the complete factored expression.

$$7\left(y+1\right)\left(y+10\right)$$

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.

$$7y^{2}+77y+70=0$$

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.

$$y=\frac{-77±\sqrt{77^{2}-4\times 7\times 70}}{2\times 7}$$

Square 77.

$$y=\frac{-77±\sqrt{5929-4\times 7\times 70}}{2\times 7}$$

Multiply -4 times 7.

$$y=\frac{-77±\sqrt{5929-28\times 70}}{2\times 7}$$

Multiply -28 times 70.

$$y=\frac{-77±\sqrt{5929-1960}}{2\times 7}$$

Add 5929 to -1960.

$$y=\frac{-77±\sqrt{3969}}{2\times 7}$$

Take the square root of 3969.

$$y=\frac{-77±63}{2\times 7}$$

Multiply 2 times 7.

$$y=\frac{-77±63}{14}$$

Now solve the equation y=\frac{-77±63}{14} when ± is plus. Add -77 to 63.

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

Divide -14 by 14.

$$y=-1$$

Now solve the equation y=\frac{-77±63}{14} when ± is minus. Subtract 63 from -77.

$$y=-\frac{140}{14}$$

Divide -140 by 14.

$$y=-10$$

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 -10 for x_{2}.

$$7y^{2}+77y+70=7\left(y-\left(-1\right)\right)\left(y-\left(-10\right)\right)$$

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

$$7y^{2}+77y+70=7\left(y+1\right)\left(y+10\right)$$

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

$$x ^ 2 +11x +10 = 0$$

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

$$r + s = -11 rs = 10$$

Two numbers r and s sum up to -11 exactly when the average of the two numbers is \frac{1}{2}*-11 = -\frac{11}{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>

$$r = -\frac{11}{2} - u s = -\frac{11}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 10

$$(-\frac{11}{2} - u) (-\frac{11}{2} + u) = 10$$

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

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

Simplify the expression by subtracting \frac{121}{4} on both sides

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

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

$$u^2 = \frac{81}{4} u = \pm\sqrt{\frac{81}{4}} = \pm \frac{9}{2}$$

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

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

Evaluate

$$7\left(y+1\right)\left(y+10\right)$$