Problem 56
Question
Solve equation by completing the square. $$ x^{2}+7 x-8=0 $$
Step-by-Step Solution
Verified Answer
The solutions to the equation \(x^{2}+7x-8=0\) are \(x = -2\) and \(x = -4.5\).
1Step 1: Rewrite The Equation
Rewrite the given equation \(x^2 + 7x - 8 = 0\) in the form \(x^2 + Bx = -C\). Here B is 7 and C is 8. So, we get the equation \(x^2 + 7x = 8\).
2Step 2: Complete The Square
To complete the square, we take half of the coefficient of x, square it and add to both sides of the equation. The coefficient of x here is 7, so half of it is \(\frac{7}{2}\) and the square of \(\frac{7}{2}\) is \(12.25\). Adding these to both sides results in \(x^2 + 7x + 12.25 = 8 + 12.25\). This simplifies to \((x + \frac{7}{2})^2 = 20.25\).
3Step 3: Square Root
To solve for x, take the square root of both sides of the equation. Remember that the square-root of a number has two values, one positive and one negative. Therefore, we get \(x + \frac{7}{2} = ± \sqrt{20.25}\), which simplifies to \(x + \frac{7}{2} = ± 4.5\).
4Step 4: Solve for x
Solve for x by subtracting \(\frac{7}{2}\) from both sides of the equation to receive \(x = - \frac{7}{2} ± 4.5\). This simplifies to the two solutions \(x = -2\) and \(x = -4.5\).
Key Concepts
Algebraic EquationsQuadratic EquationsSolving Quadratics by Completing the SquareSquare Root Method
Algebraic Equations
Algebraic equations are mathematical statements that express the equality of two algebraic expressions. These equations consist of variables, coefficients, constants, and operations such as addition, subtraction, multiplication, and division. For instance, in the equation \(x^2 + 7x - 8 = 0\), \(x\) is the variable, 7 is the coefficient of \(x\), and -8 is the constant. Solving an algebraic equation involves finding the value(s) of the variable(s) that make the equation true. In this case, we are aiming to find the value of \(x\) that satisfies the condition that the expressions on both sides of the equals sign have the same value.
When solving equations, one of the goals is to isolate the variable to one side of the equation. This involves manipulating the equation by performing the same operations on both sides of the equal sign to maintain the balance. Algebraic equations can vary in complexity, ranging from simple linear equations to more complicated polynomial ones. The strategy adopted to solve them often depends on the form and degree of the equation.
When solving equations, one of the goals is to isolate the variable to one side of the equation. This involves manipulating the equation by performing the same operations on both sides of the equal sign to maintain the balance. Algebraic equations can vary in complexity, ranging from simple linear equations to more complicated polynomial ones. The strategy adopted to solve them often depends on the form and degree of the equation.
Quadratic Equations
A quadratic equation is a type of polynomial equation of the second degree, typically in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a\) is not zero. These equations are characterized by the presence of the squared term \(x^2\), which distinguishes them from linear equations. Quadratics can have two, one, or no real solutions depending on the discriminant, which is calculated as \(b^2 - 4ac\).
The solutions to a quadratic equation can be found using various methods such as factoring, using the quadratic formula, graphing, or completing the square. The method of completing the square is particularly useful when the equation cannot be easily factored or when one desires to derive the quadratic formula. In our exercise \(x^2 + 7x - 8 = 0\), after rearranging the terms to the form \(x^2 + Bx = -C\), we move towards converting it into a perfect square trinomial, which makes the process of finding solutions easier.
The solutions to a quadratic equation can be found using various methods such as factoring, using the quadratic formula, graphing, or completing the square. The method of completing the square is particularly useful when the equation cannot be easily factored or when one desires to derive the quadratic formula. In our exercise \(x^2 + 7x - 8 = 0\), after rearranging the terms to the form \(x^2 + Bx = -C\), we move towards converting it into a perfect square trinomial, which makes the process of finding solutions easier.
Solving Quadratics by Completing the Square
Completing the square is a technique for solving quadratic equations that involves transforming the equation into a perfect square trinomial. This method is highly systematic and can be used for any quadratic equation. Here's a brief walkthrough of the method:
By efficiently utilizing the completing the square method, solving equations like \(x^2 + 7x - 8 = 0\) becomes more intuitive and less dependent on memorization of other methods such as the quadratic formula.
- Rewrite the quadratic equation so that the constant term is on the opposite side of the equation from the variable terms.
- Find half of the coefficient of the linear term (the term with \(x\)), square it, and add it to both sides of the equation. This forms a perfect square trinomial on one side.
- Rewrite the perfect square trinomial as the square of a binomial.
- Take the square root of both sides of the equation, remembering to include the plus and minus roots.
- Solve for the variable by isolating it on one side of the equation.
By efficiently utilizing the completing the square method, solving equations like \(x^2 + 7x - 8 = 0\) becomes more intuitive and less dependent on memorization of other methods such as the quadratic formula.
Square Root Method
The square root method is a part of the process of completing the square to solve quadratic equations. It comes into play after rewriting the equation as a perfect square trinomial. At this point, the next step is to isolate the squared term and take the square root of both sides, which will yield two solutions for the variable due to the fundamental property that a square root has both a positive and a negative value.
In our specific problem, after forming the perfect square trinomial \((x + \frac{7}{2})^2 = 20.25\), we apply the square root method. Taking the square root of both sides gives us \(x + \frac{7}{2} = \pm \sqrt{20.25}\), which simplifies to \(x + \frac{7}{2} = \pm 4.5\). By then isolating \(x\), we arrive at the solutions to the original equation. It's important for students to remember that when they apply the square root method, they must consider both the positive and negative square roots to find all possible solutions to the equation.
In our specific problem, after forming the perfect square trinomial \((x + \frac{7}{2})^2 = 20.25\), we apply the square root method. Taking the square root of both sides gives us \(x + \frac{7}{2} = \pm \sqrt{20.25}\), which simplifies to \(x + \frac{7}{2} = \pm 4.5\). By then isolating \(x\), we arrive at the solutions to the original equation. It's important for students to remember that when they apply the square root method, they must consider both the positive and negative square roots to find all possible solutions to the equation.
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