Predict without solving
The discriminant reveals whether a quadratic has two distinct real roots, equal real roots or no real roots without first finding them.
Learning goals
- Compute D=b^2-4ac using signed coefficients.
- Infer root nature from the sign of D.
- Determine parameters for equal roots and test existence.
Concept lesson
Coefficients carry signs
First write ax^2+bx+c=0. For 2x^2-4x+3=0, a=2, b=-4 and c=3.
Discriminant decision table
D=b^2-4ac. D>0: two distinct real roots. D=0: two equal real roots. D<0: no real roots.
Compute with a negative b
For 2x^2-4x+3=0, D=(-4)^2-4(2)(3)=16-24=-8. Brackets preserve b's sign; squaring gives positive 16.
State nature completely
Since D=-8<0, the equation has no real roots. A complete conclusion states both the comparison and nature.
Parameter for equal roots
For 2x^2+kx+3=0 to have equal roots, set k^2-24=0, so k=plus or minus 2 square root 6. Since a=2 remains non-zero, both values preserve a quadratic equation.
Sort by D sign
Compute D for three equations and place each under distinct real, equal real or no real roots. The checker records sign-calculation and decision-table errors separately.
Check the parameter result
After solving D=0, substitute every parameter back, confirm a is non-zero and recompute D.
Can the design exist?
Form the quadratic implied by a proposed dimension or location. Use D: if D<0, no real measurement satisfies the conditions; if D is non-negative, real candidates exist and still need contextual checking.
Two-mark discriminant structure
Write a,b,c with signs, substitute in D=b^2-4ac, calculate, compare D with zero and state root nature.
Exit ticket
Determine the nature of 3x^2-2x+1=0. Then find k for equal roots of x^2+kx+9=0 and verify each value.