If the quadratic equation 𝑝𝑥
squared minus two root five 𝑝𝑥 plus 15 is equal to zero has two equal roots, find
the value of 𝑝.
Remember, for a quadratic equation
of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero, we can calculate the
number of roots it has by considering the discriminant, which is given by 𝑏 squared
minus four 𝑎𝑐.
If the discriminant is greater than
zero, the equation has two distinct, thus different, real roots. If the discriminant is less than
zero, the equation has no real roots. This is because, in the quadratic
formula, we find the square root of the discriminant. And we can’t evaluate the square
root of a negative number using real numbers. If the discriminant is equal to
zero, it must have two identical roots.
If we consider the graph of a
quadratic equation, the roots represent the 𝑥-intercepts. They tell us where the graph
crosses the 𝑥-axis. If the equation has two identical
roots, it means that the vertex, the turning point of the graph, sits on the
𝑥-axis. The quadratic in our question has
two equal roots. So we know that its discriminant
must be equal to zero.
Let’s look at our equation and
identify the relevant parts of the discriminant. 𝑎 is the coefficient of 𝑥
squared. It’s 𝑝. 𝑏 is the coefficient of 𝑥. And for this equation, that’s
negative two root five multiplied by 𝑝. 𝑐 is the constant. It’s 15.
Substituting these values into our
formula for the discriminant gives us negative two root five 𝑝 all squared minus
four multiplied by 𝑝 multiplied by 15. We need to be a little bit careful
squaring this first expression.
We’ll square each part
individually. Negative two squared is negative
two multiplied by negative two. It’s four. The square root of five squared is
simplify five. And 𝑝 multiplied by 𝑝 is 𝑝
squared. Four multiplied by five multiplied
by 𝑝 squared is 20𝑝 squared. And four multiplied by 𝑝
multiplied by 15 is 60𝑝. The discriminant of this quadratic
equation then is 20𝑝 squared minus 60𝑝.
Remember, we said that, for this
equation to have two equal roots, its discriminant must be equal to zero. So we can form an equation in terms
of 𝑝 by making the value that we got for our discriminant, 20𝑝 squared minus 60𝑝,
equal to zero.
To solve this equation for 𝑝,
we’ll factorise the expression on the left-hand side. The highest common factor of both
terms in the expression for 𝑝 is 20𝑝. So when we factorise, 20𝑝 goes on
the outside of the bracket. 20𝑝 squared divided by 20𝑝 is 𝑝,
and negative 60𝑝 divided by 20𝑝 is negative three. So our equation becomes 20𝑝
multiplied by 𝑝 minus three is equal to zero.
In this expression, we have two
terms whose product is zero. That means at least one of these
terms must be zero. We say either 20𝑝 is equal to zero
or 𝑝 minus three is equal to zero. If 20𝑝 is equal to zero, 𝑝 must
be equal to zero.
We solve the second equation by
adding three to both sides. And that gives us a value of 𝑝 is
equal to three. We have two possible values of
𝑝. And we need to make a decision
whether 𝑝 is equal to zero or it’s equal to three. If 𝑝 is equal to zero, our
equation becomes zero minus zero plus 15 is equal to zero, which is simply 15 is
equal to zero. That’s meaningless. For the quadratic equation to have
two equal roots then, 𝑝 must be equal to three.