Question

Use the quadratic formula to solve the equation. $$9 x^{2}+14 x+3=0$$

Short answer

The solutions to the quadratic equation \(9x^2 + 14x + 3 = 0\) using the quadratic formula are \(x = \frac{-7 + \sqrt{22}}{9}\) and \(x = \frac{-7 - \sqrt{22}}{9}\)

Step-by-step solution

Identify the coefficients from the quadratic equation

The quadratic equation is \(9x^2 + 14x + 3 = 0\). Here, the coefficients are \(a = 9\), \(b = 14\), and \(c = 3\)

Substitute the coefficients into the quadratic formula

Substitute \(a = 9\), \(b = 14\), and \(c = 3\) into the quadratic formula, so it becomes, \[x = \frac{-14 \pm \sqrt{(14)^2 - 4*9*3}}{2*9}\]

Simplify under the square root symbol

\[x = \frac{-14 \pm \sqrt{196 - 108}}{18} = \frac{-14 \pm \sqrt{88}}{18}\]

Simplify square root

Simplify the equation \[x = \frac{-14 \pm \sqrt{(4*22)}}{18} = \frac{-14 \pm 2\sqrt{22}}{18}\]

Separate the plus and minus solutions

The solutions will be \[x = \frac{-14 + 2\sqrt{22}}{18}, x = \frac{-14 - 2\sqrt{22}}{18}\]

Simplify the solutions

Simplify the answers to get the solutions to the expressions, \[x = \frac{-7 + \sqrt{22}}{9}, x = \frac{-7 - \sqrt{22}}{9}\]

Key concepts

Solving Quadratic Equations

The process to solve a quadratic equation is essentially finding the values of the variable that make the equation true. Quadratic equations are polynomial equations of the second degree, which means the highest power of the variable, usually denoted as 'x', is two. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term.

To find the solutions, also known as 'roots', we can apply several methods, such as factoring, completing the square, or using the quadratic formula: \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). The quadratic formula is a powerful and universal tool, as it can provide solutions to any quadratic equation, even when other methods fail or are difficult to apply. It requires identifying the coefficients \(a\text{, }b\text{, and }c\), plugging them into the formula, and then simplifying the expression to get the values of \(x\).

Coefficients of Quadratic Equations

The coefficients of a quadratic equation are the numerical factors that are multiplied by the terms \( x^2 \) (the quadratic term), \( x \) (the linear term), and the constant term, which does not contain the variable. In the equation \( ax^2 + bx + c = 0 \), \( a \) is the coefficient of the quadratic term, \( b \) is the coefficient of the linear term, and \( c \) is the constant.

Importance of Coefficients

• The coefficient \( a \) determines the parabola's direction of opening: if \( a > 0 \) it opens upwards, and if \( a < 0 \) it opens downwards.
• The coefficient \( b \) affects the equation's symmetry and the location of the vertex on the x-axis.
• The constant \( c \) represents the y-intercept, the point where the parabola crosses the y-axis.

In solving a quadratic equation using the quadratic formula, the first step is to accurately identify these coefficients, as they are crucial for substituting into the formula to find the roots.

Simplifying Square Roots in Quadratic Solutions

Once the quadratic formula is applied, simplifying the square root is a common step in finding the precise solutions. The square root portion of the quadratic formula is called the discriminant, \( \sqrt{b^2 - 4ac} \). Simplifying the discriminant makes the equation easier to understand and solve.

To simplify a square root, look for factors that are perfect squares, such as \( 4 \), \( 9 \), \( 16 \), and so forth. For example, if we have \( \sqrt{88} \), we can break it down into \( \sqrt{4 \cdot 22} \). \( \sqrt{4} \) is a perfect square and equals \( 2 \) resulting in \( 2\sqrt{22} \) after simplification.

Why Simplify?

• Simplification can reveal the most 'reduced' form of the solution, making it easier to understand or further manipulate.
• It may also be necessary when comparing roots or checking the solutions against potential factors provided in multiple choice questions or real-life applications.

It's important to practice simplifying square roots to become proficient in solving quadratic equations efficiently and accurately.