Problem 75
Question
Solve the equation using any convenient method. $$5 x^{2}=3 x+1$$
Step-by-Step Solution
Verified Answer
The solutions for the equation are \(x1 = 0.2 + 0.4\sqrt{3}\) and \(x2 = 0.2 - 0.4\sqrt{3}\)
1Step 1: Rewrite the Equation
Rewrite the given equation into standard quadratic form, ax^2 + bx + c = 0. This is achieved by subtracting \(3x + 1\) from both sides to get \(5x^{2} - 3x - 1 = 0\).
2Step 2: Determine the Coefficients
Identify the coefficients \(a\), \(b\), and \(c\) in the rewritten equation. For this equation \(a = 5, b = -3\), and \(c = -1\).
3Step 3: Use the Quadratic Formula
Solve for \(x\) using the Quadratic Formula \(x = \[-b \pm \sqrt{b^{2}-4ac}\] / 2a\). Plug in the identified coefficients (a, b, and c) from step 2 into the formula, keeping in mind the importance of sign placement when using the formula. The result will be \(x = [\3 \pm \sqrt{(-3)^{2}-4(5)(-1)}] / 2*5\).
4Step 4: Simplify the equation
Simplify the obtained equation further to find the roots (or solutions) of the original equation. After simplification, the solutions are obtained as \(x1 = 0.2 + 0.4\sqrt{3}\) and \(x2 = 0.2 - 0.4\sqrt{3}\)
Key Concepts
Quadratic FormulaStandard Quadratic FormFactoring QuadraticsFinding Roots of Quadratic
Quadratic Formula
The Quadratic Formula is a powerful tool used to find the roots of quadratic equations. It is presented as \( x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \), where \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term. The beauty of this formula lies in its universality, as it can solve any quadratic equation, regardless of its form.
The formula includes a discriminant — \( b^2-4ac \) — which determines the nature of the roots. A positive discriminant suggests two real and distinct roots, zero indicates one real root (repeated), and a negative discriminant signifies two complex roots. By following the formula, we ensure that we account for both possible roots with the \( \pm \) sign, leading us to the two solutions for \( x \) that satisfy the quadratic equation.
The formula includes a discriminant — \( b^2-4ac \) — which determines the nature of the roots. A positive discriminant suggests two real and distinct roots, zero indicates one real root (repeated), and a negative discriminant signifies two complex roots. By following the formula, we ensure that we account for both possible roots with the \( \pm \) sign, leading us to the two solutions for \( x \) that satisfy the quadratic equation.
Standard Quadratic Form
Every quadratic equation can be written in the standard quadratic form \( ax^2 + bx + c = 0 \). In this form, \( a \) represents the quadratic coefficient, \( b \) is the linear coefficient, and \( c \) is the constant term. The values of \( a \) and \( b \) can be any real number, but \( a \) must not be zero to maintain the equation's quadratic nature.
Identifying this form is crucial because it sets the stage for applying solutions methods such as factoring, using the Quadratic Formula, or completing the square. It's helpful to think of this form as the quadratic equation 'template' which allows us to systematically approach solving for the roots.
Identifying this form is crucial because it sets the stage for applying solutions methods such as factoring, using the Quadratic Formula, or completing the square. It's helpful to think of this form as the quadratic equation 'template' which allows us to systematically approach solving for the roots.
Factoring Quadratics
One of the methods to solve quadratic equations is by factoring, which means rewriting the quadratic equation as a product of two binomial expressions. Factoring is based on the zero-product property, which states that if \( ab=0 \), then either \( a=0 \) or \( b=0 \), or both.
To factor a quadratic, you look for two numbers that multiply to give \( ac \) and add to give \( b \) when \( a \) is 1. If \( a \) is not 1, the process becomes slightly more complex, and you may need to employ techniques such as the AC method or other factoring strategies.
To factor a quadratic, you look for two numbers that multiply to give \( ac \) and add to give \( b \) when \( a \) is 1. If \( a \) is not 1, the process becomes slightly more complex, and you may need to employ techniques such as the AC method or other factoring strategies.
Finding Roots of Quadratic
The process of finding the roots of a quadratic equation — also known as solving the quadratic — involves determining the values of \( x \) that make the equation true. Roots are the values at which the quadratic graph crosses the \( x \) axis. They are sometimes referred to as zeros or solutions.
There are several methods to do this, each with its own benefits: factoring (as indicated above), taking square roots, completing the square, or using the Quadratic Formula. In whichever method, the principle is to rewrite the equation in such a way that applying basic algebraic principles will generate the solutions for \( x \). Understanding how to manipulate these equations and use these methods flexibly is key to mastering quadratics.
There are several methods to do this, each with its own benefits: factoring (as indicated above), taking square roots, completing the square, or using the Quadratic Formula. In whichever method, the principle is to rewrite the equation in such a way that applying basic algebraic principles will generate the solutions for \( x \). Understanding how to manipulate these equations and use these methods flexibly is key to mastering quadratics.
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