coral-application
Demerox NomelJe veux créé une application qui permet de résoudre de manière très impeccable les exercices de l'école primaire jusqu'à l'université
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Your Exercise
Solve for x: 2x² - 5x + 3 = 0. Give exact values and verify your solutions by substitution.
Solution Navigator
- ✓Step 1Identify the unknowns
- ✓Step 2Write the standard form
- Step 3Calculate the discriminant
- Step 4Find the two roots
- Step 5Verify the solutions
- Step 6State the final answer
Begin by reading the problem carefully and identifying all unknown quantities. In this quadratic equation, we have one unknown — x — which represents the value(s) satisfying the equation. Assign a clear variable name and note any constraints (e.g., x must be a real number, or x > 0 for physical problems).
Formulas & Rules Explained
All mathematical rules and formulas referenced in this solution
Used to find the roots (solutions) of any quadratic equation in the standard form ax² + bx + c = 0. The ± symbol means there are two possible solutions.
The discriminant determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 means no real roots (complex solutions).
Vieta's formulas relate the coefficients of a quadratic to the sum and product of its roots. Useful for checking your solutions or finding roots without fully solving the equation.
Rewriting a quadratic in vertex form by creating a perfect square trinomial. Reveals the vertex (−b/2a, −Δ/4a) and is the method from which the quadratic formula is derived.
Understanding the Concept
Key Concept
A quadratic equation takes the standard form ax² + bx + c = 0. It represents a parabola when graphed and arises in countless real-world contexts — from projectile motion to optimisation problems. Every quadratic has exactly two roots (counting multiplicity), which may be real or complex, found using the quadratic formula, factorisation, or completing the square.
Common Mistakes
- Forgetting both solutionsA quadratic always has two roots. Students often stop after finding the first value of x and miss the second solution entirely.
- Sign errors when rearrangingMoving terms across the equals sign requires flipping the sign. A common slip is keeping the original sign, leading to an incorrect standard form.
- Misapplying the discriminantWhen Δ < 0, students sometimes still attempt to find real roots. Remember: a negative discriminant means no real solutions exist.
Practice More Like This
Reinforce your understanding with similar exercises
Solve the quadratic equation 2x² − 5x + 3 = 0 using the quadratic formula and verify your solutions by substitution.
Factor the expression x² − 7x + 12 completely, then find the roots of the corresponding equation.
A projectile is launched at 30° above the horizontal with initial velocity 20 m/s. Find the maximum height reached.
Complete the square for the expression x² + 8x − 5 and use the result to identify the vertex of the parabola.
Solve the quadratic equation 2x² − 5x + 3 = 0 using the quadratic formula and verify your solutions by substitution.
Factor the expression x² − 7x + 12 completely, then find the roots of the corresponding equation.
A projectile is launched at 30° above the horizontal with initial velocity 20 m/s. Find the maximum height reached.
Complete the square for the expression x² + 8x − 5 and use the result to identify the vertex of the parabola.
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