Video Transcript
Solve the equation four 𝑥 squared
plus 40𝑥 plus 40 equals negative 60 by factoring.
We’ve been given a quadratic
equation and asked to solve it by factoring. Before we can do this, we need to
collect every term on the same side of the equation. We’ll begin by adding 60 to each
side of the equation. Doing so gives the equivalent
quadratic equation four 𝑥 squared plus 40𝑥 plus 100 equals zero.
Before attempting to factor, we can
observe that every coefficient in the equation is a multiple of four. Hence, we can simplify by dividing
the entire equation by four. Zero divided by four is still
zero. So, the right-hand side is
unchanged, but the left-hand side becomes 𝑥 squared plus 10𝑥 plus 25. This is easier to work with because
the coefficients are smaller. As the coefficient of 𝑥 squared in
this simplified equation is one, and we’re told to solve by factoring, we know that
the expression on the left-hand side can be written in the form 𝑥 plus 𝑎
multiplied by 𝑥 plus 𝑏, where 𝑎 and 𝑏 are constants that we need to
determine.
The values 𝑎 and 𝑏 must satisfy
two properties: their sum must be equal to the coefficient of 𝑥 in the quadratic
equation, which is 10, and their product must be equal to the constant term, which
is 25. With a bit of thought, or perhaps
by listing the factor pairs of 25, we identify that the values of 𝑎 and 𝑏 are both
five. Hence, the factored form of the
quadratic equation is 𝑥 plus five multiplied by 𝑥 plus five is equal to zero.
So, we’ve written this quadratic
equation in its fully factored form. In order to now solve the equation,
we recall that if the product of two values is equal to zero, then at least one of
those values must themselves be equal to zero. So, we set each of the two factors
equal to zero and then solve the resulting equations. In this case, we have a repeated
factor of 𝑥 plus five, and so we actually only need to solve this equation
once. Subtracting five from each side of
this equation gives 𝑥 equals negative five. The solution to the quadratic
equation is therefore 𝑥 equals negative five.
A slightly alternative approach
would have been to write the quadratic equation in the form 𝑥 plus five all squared
equals zero. We would then argue that if
something squared is equal to zero, that something must be equal to the square root
of zero, which is also zero. So, we have 𝑥 plus five equals
zero which leads to 𝑥 equals negative five as before. We can check our solution is
correct by substituting this value of 𝑥 back into the original equation. Doing so gives 100 minus 200 plus
40, which is equal to negative 60 as required.
We’ve found that the solution to
the equation four 𝑥 squared plus 40𝑥 plus 40 equals negative 60 is 𝑥 equals
negative five.
