QUADRATIC EQUATIONS
Frequently Asked Questions
A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\) where \(a,\ b\, c\) are real numbers and \(a \neq 0\).
If \(a = 0\), the equation becomes linear and no longer contains a squared term, so it cannot be quadratic.
The standard form is \(ax^2 + bx + c = 0\).
The word “quadratic” comes from “quad,” meaning square, because the highest power of the variable is 2.
The solutions of \(ax^2 + bx + c = 0\) are \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
The discriminant \(D\) is the expression \(b^2 - 4ac\) found inside the square root of the quadratic formula.
It indicates two distinct real roots.
It indicates one real and repeated root.
It indicates no real roots; the solutions are complex.
By splitting the middle term into two terms whose product is (ac), factoring the expression, and using the zero-product property.
If \(pq = 0\), then either \(p = 0\) or \(q = 0\). It is used to solve factored quadratic equations.
It means expressing \(bx\) as the sum of two terms whose product equals \(ac\), helping in factorization.
It is a method of rewriting a quadratic as a perfect square expression to solve the equation.
It helps derive the quadratic formula and solve equations that are not easy to factor.
Ensure \(a = 1\), take half of the coefficient of \(x\), square it, add it to both sides, form a perfect square, and solve.
Roots are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\).
For equation \(ax^2 + bx + c = 0\): sum of roots = \(-b/a\); product of roots = \(c/a\).
The roots are the \(x\)-intercepts where the parabola \(y = ax^2 + bx + c\) crosses the \(x\)-axis.
When \(D = 0\); the parabola is tangent to the x-axis.
When \(D < 0\); the graph does not cross or touch the \(x\)-axis.
It works for all types of quadratic equations, even when factorization is difficult.
They appear in geometry, projectile motion, business profit problems, age problems, and number-based puzzles.
Shape-based problems such as area, diagonal relations, and dimensions often result in a quadratic equation.
They are used in motion under gravity, height-time relations, and projectile trajectories.
By completing the square on the general form \(ax^2 + bx + c = 0\).
An expression like \(x^2 + 2px + p^2 = (x + p)^2\).
When its discriminant is a perfect square or when integers exist that multiply to \(ac\) and sum to \(b\).
Using relations involving area, speed, number constraints, or algebraic identities to form \(ax^2 + bx + c = 0\).
If length = breadth + 3 and area = 40 sq units, then \(b(b+3) = 40\) becomes a quadratic equation.
A quadratic equation without a linear term, i.e., of the form \(ax^2 + c = 0\).
Quadratic equations that contain all three terms: \(ax^2\), \(bx\), and \(c\).
A quadratic equation where \(a = 1\), e.g., \(x^2 + 5x + 6 = 0\).
Substituting answers back ensures the solution satisfies the original problem context.
Factorization-based questions, quadratic formula problems, word problems, discriminant evaluation, and root nature analysis.
Factorization is fastest when applicable; otherwise, the quadratic formula is the safest and most reliable.
Because the discriminant becomes negative, making the square root of a negative number impossible in real numbers.
Forgetting to bring the equation to standard form before applying methods or miscalculating the discriminant.
The sign determines the curve orientation and affects nature of roots.
Rearranging and simplifying the equation so that all terms are on one side of the equal sign.
Yes, when the discriminant is not a perfect square.
Yes, when \(D\) is positive but not a perfect square.
\(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola.
Only basic understanding; detailed graphing is taught in higher classes.
Maximizing area of a rectangular garden using fixed fencing length leads to a quadratic equation.
It quickly determines the nature of roots without solving the full equation.
Problems involving numbers, age, geometry, motion, mixtures, and profit that reduce to quadratic equations.
Because methods like factorization or formula application work only in standard form.
Practice factorization, memorize formulas, and solve multiple word problems to gain confidence.
Substitute them in the original equation and verify if both sides balance.
\(ax^2 + bx + c = 0\) without common factors and with simplified coefficients.
