SAT quadratics: use the discriminant to count real solutions
The expression b² − 4ac tells you whether a quadratic crosses the x-axis twice, touches it once, or misses it.
SPSAT Practice Editorial5 min read
The short version
What you'll take away
- Put the equation in ax² + bx + c = 0 form before identifying a, b, and c.
- A positive discriminant means two distinct real solutions; zero means one; negative means none.
- A zero discriminant means the parabola touches the x-axis at its vertex.
In this article
Count solutions without solving the whole equation
For \(\displaystyle ax^2+bx+c=0\), with \(\displaystyle a\ne0\), compute the discriminant \(\displaystyle D=b^2-4ac\). The quadratic formula divides \(\displaystyle -b\pm\sqrt{D}\) by \(\displaystyle 2a\). If D is positive, the plus and minus give two distinct real answers. If D is zero, they give the same answer. If D is negative, its square root is not real.
For \(\displaystyle x^2-6x+9=0\), D is \(\displaystyle (-6)^2-4(1)(9)=0\). The equation is also \(\displaystyle (x-3)^2=0\), so x = 3 is the only real solution.
See the skill
What the sign of b² − 4ac says
D > 0
Two distinct real solutions; the parabola crosses the x-axis twice.
D = 0
One distinct real solution; the parabola touches the x-axis once.
D < 0
No real solutions; the parabola does not reach the x-axis.
Use a parameter to make exactly one solution
Suppose \(\displaystyle x^2-4x+k=0\) must have exactly one real solution. Here a = 1, b = −4, and c = k. Set the discriminant to zero: \(\displaystyle 16-4k=0\), so \(\displaystyle k=4\). Checking gives \(\displaystyle x^2-4x+4=(x-2)^2\), which touches zero at x = 2.
If an equation is not equal to zero, move every term to one side first. For example, \(\displaystyle x^2+3x=7\) has c = −7, not 7.
Check your discriminant
You only need the sign of D to answer this question.
Your turn
Quick practice
How many distinct real solutions does 2x² + 3x + 5 = 0 have?
Hint: Identify a = 2, b = 3, and c = 5.
Sources & further reading
One step from here
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