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Learning intentions:
In this section we will examine:
- The discriminant and its applications in solving quadratic equations.
The discriminant
The discriminant (Δ) comes from the expression under the square root sign in the general quadratic formula:
- The values of a, b and c correspond to the coefficients in the general quadratic equation (ax2 + bx + c = 0).
Please note:
- If the discriminant (Δ) is a square number (i.e. 1, 4, 9, 16, 25, ...) then the solutions will be rational (able to be expressed as a fraction).
Determining the number of solutions for a quadratic
By identifying whether Δ is positive, negative or zero, we can determine how many solutions (or x-intercepts) there are.
Two solutions
- When the discriminant is positive (Δ > 0), two solutions (x-intercepts) exist.
One solution
- When the discriminant is zero (Δ = 0), one solution (x-intercept) exists.
No solutions
- When the discriminant is negative (Δ < 0), no solutions (x-intercepts) exist.
Also refer to solving simultaneous equations involving quadratics.
Figure 1 - (L - R) Quadratics with zero, one and two solutions (x-intercepts).
4E - VIDEO EXAMPLE 1:
Use the discriminant to determine how many solutions exist for the following:
a)
b)
c)
4E - VIDEO EXAMPLE 2:
Use the discriminant to show that the quadratic y = -x2 + (m + 2)x + 3 - 3m has rational x-intercepts for any value of m.
Success criteria:
You will be successful if you can:
- Use the discriminant to determine the number of solutions for a quadratic equation.
- Use the discriminant to solve problems involving the intersection between two curves.
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