Problem 24 - Algebra 2 and Trigonometry

Question

One of the roots is given. Find the other root. \(m^{2}-4 m+n=0 ; 3\)

Step-by-Step Solution

Answer

The other root is 1.

Step 1: Identify the given root

The problem states that one of the roots of the equation is 3. This means that substituting 3 for \(m\) in the equation \(m^{2}-4m+n=0\) should satisfy it.

Step 2: Substitute the known root into the equation

Substitute the value 3 into the equation: \((3)^2 - 4(3) + n = 0\). Simplifying, we get \(9 - 12 + n = 0\).

Step 3: Solve for the constant term \(n\)

From \(9 - 12 + n = 0\), we get \(-3 + n = 0\). Solving for \(n\) gives us \(n = 3\).

Step 4: Write the quadratic equation with the found \(n\)

With \(n = 3\), the quadratic equation becomes \(m^2 - 4m + 3 = 0\).

Step 5: Use the quadratic formula to find the other root

The quadratic formula is given by \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b = -4\), and \(c = 3\). Substitute these values into the formula: \(m = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)}\).

Step 6: Calculate the discriminant

The discriminant \(b^2 - 4ac\) is \((-4)^2 - 4(1)(3) = 16 - 12 = 4\).

Step 7: Find the roots using the quadratic formula

Using the quadratic formula, we get \(m = \frac{4 \pm \sqrt{4}}{2}\). This simplifies to \(m = \frac{4 \pm 2}{2}\). Thus, the roots are \(m = 3\) and \(m = 1\).

Key Concepts

Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

To solve for \( x \), you substitute the values of \( a \), \( b \), and \( c \) from the equation into the quadratic formula.
The symbol \( \pm \) indicates that there are typically two solutions: one involving the addition of \( \sqrt{b^2 - 4ac} \) and one involving its subtraction. This is because a quadratic equation represents a parabola, which can intersect the x-axis at two points, or not at all, depending on the nature of the discriminant.

Exploring the Discriminant

The discriminant is a specific part of the quadratic formula given by \( b^2 - 4ac \). It plays a crucial role in determining the nature and number of roots for a quadratic equation.
Here's what different values of the discriminant mean:

• If \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
• If \( b^2 - 4ac = 0 \), there is exactly one real root. This happens when the parabola touches the x-axis at a single point (it is tangent to the x-axis).
• If \( b^2 - 4ac < 0 \), the equation does not have real roots. Instead, it has two complex conjugate roots, indicating the parabola does not intersect the x-axis at all.

Understanding the discriminant helps you predict how many solutions you will get before actually solving the equation.

Finding the Roots of Equations

The roots of a quadratic equation are the solutions to the equation \( ax^2 + bx + c = 0 \). These values of \( x \) make the equation true—to graphically interpret, they are the points where the parabola intersects the x-axis.
Using the quadratic formula, you can find these roots by:

• Calculating the discriminant \( b^2 - 4ac \) first.
• Substituting the values of \( a \), \( b \), and \( c \) into the quadratic formula.
• Solving for \( x \) gives you two potential roots: \( \frac{-b + \sqrt{b^2 - 4ac}}{2a} \) and \( \frac{-b - \sqrt{b^2 - 4ac}}{2a} \).

It is possible for a quadratic equation to have one, two, or no real roots—depending largely on the discriminant value. For equations with real roots, you will typically see two solutions unless the discriminant is zero, in which case the roots are repeated (a perfect square).