Question

Which of these is the correct quadratic formula for equations in the form \(a x^{2}+b x+c=0\) ? (A) \(x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\) OR (B) \(x=\frac{b \pm \sqrt{b^{2}+4 a c}}{a}\)

Short answer

The correct quadratic formula is (A) \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step-by-step solution

Identify the Standard Form of a Quadratic Equation

A quadratic equation must be in the form \(ax^2 + bx + c = 0\). This form is crucial because the quadratic formula is derived from it.

Recall the Quadratic Formula

The standard quadratic formula used to find the roots of the equation \(ax^2 + bx + c = 0\) is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula allows us to calculate the values of \(x\) that satisfy the quadratic equation.

Compare Given Options to the Standard Formula

Compare option (A) \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) and option (B) \(x = \frac{b \pm \sqrt{b^2 + 4ac}}{a}\) with the standard quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Identify the Correct Formula

Option (A) matches the standard quadratic formula exactly, with the correct placement of \(-b\), the \(\pm\) symbol, the discriminant \(b^2 - 4ac\), and division by \(2a\). Option (B) does not match due to the wrong signs and structure.

Key concepts

Standard Form of Quadratic Equation

A quadratic equation is a polynomial equation of degree two. The standard form of a quadratic equation is written as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This form is foundational because it clearly delineates the coefficients of the equation, making calculation more straightforward and effective.

To break it down:

• \(a\) is the coefficient of \(x^2\), and it shapes the parabola's direction; if \(a\) is positive, the parabola opens upwards, and if negative, it opens downwards.
• \(b\) is the coefficient of \(x\), which affects the symmetry and orientation of the parabola.
• \(c\) is the constant term, influencing the vertical position of the parabola on the graph.

Getting equations into this form is essential before applying the quadratic formula, completing the square, or factoring.

Discriminant in Quadratic Equation

The discriminant is a crucial part of the quadratic formula, captured in the expression \(b^2 - 4ac\). It not only appears in the quadratic formula but also provides vital information about the nature of the roots of the quadratic equation.

The discriminant helps to determine:

• **Number of Real Roots**: If \(b^2 - 4ac > 0\), there are two distinct real roots. If it equals zero, there is exactly one real root, often referred to as a repeated or double root. If \(b^2 - 4ac < 0\), there are no real roots; instead, the roots are complex or imaginary numbers.
• **Type of Solutions**: With the discriminant, you can predict if the solutions are rational or irrational. A perfect square discriminant leads to rational roots, while a non-perfect square results in irrational roots.

Understanding the discriminant is immensely useful for anticipating how the quadratic will behave, without needing to solve the entire equation.

Roots of a Quadratic Equation

The roots, or solutions, of a quadratic equation are the values of \(x\) which satisfy the equation \(ax^2 + bx + c = 0\). An invaluable tool for finding these roots is the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

This formula uses:

• **Coefficients and Constant**: \(-b\), \(b^2 - 4ac\) (discriminant), and \(2a\).
• **"\(\pm\)" Symbol**: Indicates two solutions are typically possible, resulting from the plus or minus sign.

After solving for the roots, you can understand where the parabola formed by the equation crosses the x-axis. Depending on the discriminant, the parabola might cross in two places, just touch it, or not intersect it at all. Exploring roots with the quadratic formula offers comprehensive insights into quadratic relationships and their graphical interpretations.