Read the Real-Root Count from a Discriminant
The discriminant D=B²−4AC is the expression under the formula’s square root.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: How many real roots does x²+4x+3=0 have? Candidate answers: (A) Every real number is a root (B) No real roots (C) One repeated real root (D) Two distinct real roots.
- Independent check inputs: How many real roots does x²+6x+9=0 have? Candidate answers: (A) Every real number is a root (B) No real roots (C) One repeated real root (D) Two distinct real roots.
Learn the idea
Quadratic equations and graphs express the same underlying structure. A discriminant predicts real-root count, the quadratic formula calculates roots, and vertex form displays shifts, direction and stretching directly. The discriminant D=B²−4AC is the expression under the formula’s square root. Its sign distinguishes two distinct real roots, one repeated real root, or no real roots.
Here is the method on the supplied teaching case. D=(4)²−4(3)=4. A positive discriminant gives two distinct real roots; zero gives one repeated real root; a negative discriminant gives no real roots. Here the answer is: two distinct real roots.
A negative discriminant means no real roots, not no complex roots. State the number system requested; a zero discriminant repeats one root rather than producing two distinct real solutions.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: How many real roots does x²+4x+3=0 have? Candidate answers: (A) Every real number is a root (B) No real roots (C) One repeated real root (D) Two distinct real roots. Solution: D=(4)²−4(3)=4. A positive discriminant gives two distinct real roots; zero gives one repeated real root; a negative discriminant gives no real roots. Here the answer is: two distinct real roots. The final answer is Two distinct real roots. A negative discriminant means no real roots, not no complex roots. State the number system requested; a zero discriminant repeats one root rather than producing two distinct real solutions.
Check your reasoning
Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.
Helpful tips
- The discriminant D=B²−4AC is the expression under the formula’s square root.
- A negative discriminant means no real roots, not no complex roots.
See the answer and explanation
One repeated real root
D=(6)²−4(9)=0. A positive discriminant gives two distinct real roots; zero gives one repeated real root; a negative discriminant gives no real roots. Here the answer is: one repeated real root.
Take it further
Changed-input task: How many real roots does x²+18x+83=0 have? Candidate answers: (A) Every real number is a root (B) No real roots (C) One repeated real root (D) Two distinct real roots. Show the mathematical reason, not only the selected answer. Expected result: No real roots. D=(18)²−4(83)=-8. A positive discriminant gives two distinct real roots; zero gives one repeated real root; a negative discriminant gives no real roots. Here the answer is: no real roots.
Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.
More practice: try new cases
Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.
Your choices are temporary. Feedback appears after you check all the answers.
Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.
Check the subject sources
These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.