Problem 54 - Algebra 1

Question

Solve the equation. $$3 x^{2}+11 x+10=0$$

Step-by-Step Solution

Answer

The solutions for the equation \(3x^{2}+11x+10=0\) are \(x_{1}=-5/3\) and \(x_{2}=-2\).

Step 1 - Identify the coefficients

In the quadratic equation, the general form is \(ax^{2}+bx+c=0\). Therefore, comparing our given equation with general form, we can safely say that \(a=3\), \(b=11\), and \(c=10\).

Step 2 - Apply the quadratic formula

Now we use the quadratic formula, which is \[x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\] We substitute our coefficients values (\(a=3\), \(b=11\), \(c=10\)) into the formula. This will then become: \[x=\frac{-11\pm\sqrt{11^{2}-4(3)(10)}}{2(3)}\] simplifying under the square root we get: \[x=\frac{-11\pm\sqrt{121-120}}{6}\]

Step 3 - Solve for \(x\)

Continuing the calculation of \(x\) from step 2, we get: \[x=\frac{-11\pm\sqrt{1}}{6}\] This will yield two solutions, one for the plus and one for the minus: \(x_{1}=\frac{-11+1}{6}=-5/3\) and \(x_{2}=\frac{-11-1}{6}=-2\)

Key Concepts

Quadratic Formula

The quadratic formula is a critical tool for solving polynomial equations of the second degree, commonly referred to as quadratic equations. The formula is given as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]It's designed to find the solutions, or 'roots', of the quadratic equation format \(ax^2 + bx + c = 0\). This formula is beneficial because it works for any quadratic equation, even when factoring the equation seems challenging.
To utilize this formula, you must accurately identify your equation's coefficients, which we'll explore more next. Understanding the application of each part of the formula is crucial; \(-b\) shifts the equation, while the square root expression \(\sqrt{b^2 - 4ac}\) determines if the roots are real or complex. The \(\pm\) symbol reveals the two possible roots derived from the addition and subtraction of the square root result.

Coefficient Identification

In any quadratic equation of the format \(ax^2 + bx + c = 0\), the coefficients \(a\), \(b\), and \(c\) play vital roles. These coefficients help define the equation’s unique characteristics and set the stage for solving it with the quadratic formula.
Identifying these coefficients is the crucial first step in using the quadratic formula effectively.

• \(a\) is the coefficient of \(x^2\). It's an essential part because it defines the parabola's "opening." If \(a\) is positive, the parabola opens upwards, and if negative, it opens downwards.
• \(b\) is the coefficient of \(x\). This impacts the parabola’s direction and position horizontally.
• \(c\) is the constant term in the equation. This coefficient places the vertex of the parabola along the vertical axis.

In our specific equation example \(3x^2 + 11x + 10 = 0\), by comparing with the standard form, we find \(a = 3\), \(b = 11\), and \(c = 10\). This sets up our next steps where we input these values into the quadratic formula.

Solving Quadratic Equations

To solve quadratic equations, we employ the quadratic formula after identifying the correct coefficients. Here's a step-by-step process explained using our example equation \(3x^2 + 11x + 10 = 0\).
Firstly, insert the coefficients into the formula: \[x = \frac{-11 \pm \sqrt{11^2 - 4(3)(10)}}{2(3)} \]
We perform calculations under the square root:

• Calculate \(11^2 = 121\).
• Compute \(4 \times 3 \times 10 = 120\).
• This gives us \(\sqrt{121 - 120} = \sqrt{1} = 1\).

Substituting back, we solve:

• For \(x_1 = \frac{-11 + 1}{6} = -\frac{10}{6} = -\frac{5}{3}\).
• For \(x_2 = \frac{-11 - 1}{6} = -\frac{12}{6} = -2\).

Thus, the roots of the quadratic equation \(3x^2 + 11x + 10 = 0\) are \(x = -\frac{5}{3}\) and \(x = -2\). Both roots are real and can be checked by substituting back into the original equation, affirming their correctness.