Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, typically in the form ax² + bx + c = 0, where a, b, and c are real numbers, and a ≠ 0. It represents a parabola when graphed on a coordinate plane. For example, 2x² + 3x - 5 = 0 is a quadratic equation.

To solve a quadratic equation by factorisation, first express it in the standard form ax² + bx + c = 0. Then, factorise the quadratic expression into two binomials. Set each binomial equal to zero and solve for x. For example, x² - 5x + 6 = 0 factors to (x-2)(x-3)=0, giving roots x=2 and x=3.

The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a), used to find the roots of any quadratic equation ax² + bx + c = 0. The term under the square root, b² - 4ac, is called the discriminant. For example, for 2x² + 4x - 6 = 0, the roots are x = 1 and x = -3.

The discriminant, D = b² - 4ac, indicates the nature of the roots. If D > 0, there are two distinct real roots. If D = 0, there is one real root (a repeated root). If D < 0, there are no real roots, but two complex roots. For example, in x² - 4x + 4 = 0, D=0, so there's one real root, x=2.

No, a quadratic equation cannot have more than two roots. By the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n roots (real or complex). Since a quadratic equation is degree 2, it has exactly two roots.

If the roots are α and β, the quadratic equation can be formed as x² - (α+β)x + αβ = 0. For example, if roots are 3 and -2, the equation is x² - (3-2)x + (3)(-2) = 0, which simplifies to x² - x - 6 = 0.

For a quadratic equation ax² + bx + c = 0, the sum of the roots (α+β) is -b/a, and the product (αβ) is c/a. These relationships are useful for forming equations and solving problems without explicitly finding the roots. For example, in 3x² - 6x + 2 = 0, α+β = 2 and αβ = 2/3.

Quadratic equations model various real-life scenarios, such as projectile motion, area optimization, and profit maximization. For instance, if a ball is thrown upward, its height over time can be modeled by a quadratic equation, helping to determine maximum height or time to hit the ground.

Completing the square transforms a quadratic equation into the form (x + p)² = q, making it easier to solve. This involves rearranging the equation, adding and subtracting a suitable term to form a perfect square trinomial. For example, x² + 6x + 5 = 0 becomes (x + 3)² = 4 after completing the square.

If a=0, the equation reduces to bx + c = 0, which is linear, not quadratic. The term 'quadratic' comes from 'quadratus', meaning square, referring to the x² term. Thus, a must be non-zero to maintain the equation's quadratic nature.

Substitute the value into the equation in place of x. If the equation holds true (equals zero), the value is a root. For example, to check if x=2 is a root of x² - 5x + 6 = 0, substitute: 4 - 10 + 6 = 0, confirming it is a root.

A quadratic equation has real and equal roots if its discriminant (b² - 4ac) is zero. This means the parabola touches the x-axis at exactly one point. For example, x² - 4x + 4 = 0 has a discriminant of 0, giving one real root, x=2.

No, if a quadratic equation has real coefficients, complex roots must come in conjugate pairs. Thus, it's impossible to have one real and one complex root; it's either two real roots or two complex roots.

The vertex of the parabola represents the minimum (if a > 0) or maximum (if a < 0) value. The x-coordinate of the vertex is -b/(2a), and substituting this into the equation gives the y-coordinate. For example, for f(x) = 2x² - 4x + 1, the minimum is at x=1, f(1)=-1.

A linear equation is of the first degree (e.g., ax + b = 0) and graphs as a straight line, while a quadratic equation is of the second degree (e.g., ax² + bx + c = 0) and graphs as a parabola. Linear equations have one root, whereas quadratic equations have two roots.

By calculating the discriminant (D = b² - 4ac). If D > 0, two distinct real roots; D = 0, one real root; D < 0, no real roots. For example, for x² + x + 1 = 0, D = -3, indicating no real roots.

Common mistakes include forgetting to set the equation to zero before solving, incorrect factorisation, misapplying the quadratic formula, and ignoring the ± sign when taking square roots. Always double-check calculations and ensure the equation is in standard form first.

Quadratic equations often arise in problems involving areas, such as finding the dimensions of a rectangle given its area and perimeter. For example, if a rectangle's length is twice its width and area is 50 m², setting width as x leads to the equation 2x² = 50.

The constant term 'c' affects the y-intercept of the parabola and the product of the roots (αβ = c/a). Changing 'c' shifts the parabola up or down without altering its shape. For example, in y = x² + 3x + c, varying 'c' moves the graph vertically.

Multiply every term by the least common denominator (LCD) to eliminate fractions, then solve as usual. For example, (1/2)x² + (3/4)x - 1 = 0 becomes 2x² + 3x - 4 = 0 after multiplying by 4, making it easier to solve.

The vertex is the highest or lowest point on the parabola, representing the function's maximum or minimum value. Its coordinates (-b/2a, f(-b/2a)) are crucial for graphing and optimization problems. For instance, in projectile motion, the vertex gives the maximum height reached.

No, only those that can be expressed as a product of linear factors with rational coefficients can be solved by factorisation. Others may require the quadratic formula or completing the square. For example, x² + x - 1 = 0 is best solved using the quadratic formula.

The coefficient 'a' determines the parabola's width and direction. If |a| increases, the parabola narrows; if |a| decreases, it widens. If a > 0, it opens upwards; if a < 0, it opens downwards. For example, y = 3x² is narrower than y = x².

Quadratic equations model phenomena like projectile motion, where the path is parabolic, and optics, involving reflective properties of parabolas. They also appear in kinematics equations under constant acceleration, helping calculate displacement over time.

Use the quadratic formula, as it reliably provides roots regardless of their nature. For example, x² - 2x - 1 = 0 has roots x = 1 ± √2, which are irrational. The formula ensures accurate solutions without needing factorisation.