Problem 40
Question
Solve the quadratic equation using any convenient method. $$ 3 x^{2}-16=38 $$
Step-by-Step Solution
Verified Answer
The solutions to the equations are \(x = \sqrt{18}\) and \(x = -\sqrt{18}\)
1Step 1: Rearrange the Equation
Firstly, arrange the equation correctly by moving all terms to one side of the equation. The equation becomes \(3x^{2} - 16 - 38 = 0\), which simplifies to \(3x^{2} - 54 = 0\)
2Step 2: Simplify the Equation
Next, make the equation simpler to make it easier to apply the quadratic formula. By dividing every term by 3, we get \(x^{2} - 18 = 0\)
3Step 3: Apply Quadratic Formula
Next, use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where in our equation a = 1, b = 0, c = -18. Substituting these values in the formula gives two possible values for x: \(x = \sqrt{18}\) and \(x = -\sqrt{18}\)
Key Concepts
Solving Quadratic EquationsQuadratic FormulaAlgebraic Expressions
Solving Quadratic Equations
Understanding the process of solving quadratic equations is a fundamental skill in algebra. A quadratic equation can be recognized by its standard form, which is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(a \eq 0\). When faced with a quadratic equation, such as \(3x^{2}-16=38\), the first objective is to rearrange it into standard form.
One method to solve for \(x\) after achieving standard form is factoring, provided that the equation can be easily factored into two binomial expressions. Another method is completing the square, which involves manipulating the equation to form a perfect square trinomial. However, when these methods are not viable or efficient, we resort to the quadratic formula.
The quadratic formula offers a straightforward solution by substituting the coefficients into the formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). It always works regardless of the coefficients involved, making it a universal tool for solving quadratic equations. By applying this systematic approach, one ensures that all real and complex roots of the equation will be found.
One method to solve for \(x\) after achieving standard form is factoring, provided that the equation can be easily factored into two binomial expressions. Another method is completing the square, which involves manipulating the equation to form a perfect square trinomial. However, when these methods are not viable or efficient, we resort to the quadratic formula.
The quadratic formula offers a straightforward solution by substituting the coefficients into the formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). It always works regardless of the coefficients involved, making it a universal tool for solving quadratic equations. By applying this systematic approach, one ensures that all real and complex roots of the equation will be found.
Quadratic Formula
The quadratic formula is a powerful weapon in an algebra student's arsenal. Written as \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), it is derived from the process of completing the square on the general quadratic equation. The formula provides the roots of any quadratic equation effortlessly by substituting the numerical values of \(a\), \(b\), and \(c\) into it.
Let's dissect it further: the expresssion \(b^{2} - 4ac\), known as the discriminant, determines the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it's zero, there's exactly one real root, also known as a repeated or double root. A negative discriminant indicates that the equation has complex roots.
Using the quadratic formula carries an important iterative process: calculate the discriminant, determine the square root, and finally compute the two results from the plus-minus operation. This process ensures that no real roots are left undiscovered and is applicable for all types of quadratic equations, whether they're easily factorable or not.
Let's dissect it further: the expresssion \(b^{2} - 4ac\), known as the discriminant, determines the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it's zero, there's exactly one real root, also known as a repeated or double root. A negative discriminant indicates that the equation has complex roots.
Using the quadratic formula carries an important iterative process: calculate the discriminant, determine the square root, and finally compute the two results from the plus-minus operation. This process ensures that no real roots are left undiscovered and is applicable for all types of quadratic equations, whether they're easily factorable or not.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They provide a way to represent various mathematical relationships and scenarios in a symbolic form. For instance, a quadratic expression like \(3x^{2} - 54\) conveys information about a parabolic relationship between \(x\) and the expression's value.
Understanding algebraic expressions is paramount in solving quadratic equations, as these expressions define the structure of the equation. Simplifying expressions is often the preliminary step in problem-solving, such as dividing the entire equation by 3 to reduce \(3x^2 - 54 = 0\) to \(x^2 - 18 = 0\). This simplification can make subsequent steps, like applying the quadratic formula, much easier.
Recognizing different forms of algebraic expressions, such as expanded, factored, or vertex form, can provide insights into the most suitable strategy for solving equations. By becoming familiar with various forms and transformations, students can enhance their mathematical acumen and tackle diverse algebraic challenges with greater confidence and agility.
Understanding algebraic expressions is paramount in solving quadratic equations, as these expressions define the structure of the equation. Simplifying expressions is often the preliminary step in problem-solving, such as dividing the entire equation by 3 to reduce \(3x^2 - 54 = 0\) to \(x^2 - 18 = 0\). This simplification can make subsequent steps, like applying the quadratic formula, much easier.
Recognizing different forms of algebraic expressions, such as expanded, factored, or vertex form, can provide insights into the most suitable strategy for solving equations. By becoming familiar with various forms and transformations, students can enhance their mathematical acumen and tackle diverse algebraic challenges with greater confidence and agility.
Other exercises in this chapter
Problem 40
Solve the inequality. Then graph the solution set on the real number line. \(3 x+1 \geq 2+x\)
View solution Problem 40
Find the real solution(s) of the equation involving rational exponents. Check your solutions. \(\left(x^{2}-x-22\right)^{4 / 3}=16\)
View solution Problem 40
Solve the quadratic equation by extracting square roots. List both the exact answer and a decimal answer that has been rounded to two decimal places. $$ 2 x^{2}
View solution Problem 40
Salary You accept a new job with a starting salary of \(\$ 48,000\). You receive a \(4 \%\) raise at the start of your second year, a \(5.5 \%\) raise at the st
View solution