Problem 54
Question
Write the quadratic equation in standard form. Then solve using the quadratic formula. $$-2+x^{2}=-x$$
Step-by-Step Solution
Verified Answer
The solutions to the quadratic equation are \(x=1\) and \(x=-2\).
1Step 1: Write the Quadratic Equation in Standard Form
Transform the given equation \(-2+x^{2}=-x\) into standard form by moving all terms to one side of the equation. This results in \(x^{2}+x-2=0\).
2Step 2: Identify the Coefficients
The coefficients from the standard form, \(ax^2+bx+c=0\), are a=1, b=1 and c=-2.
3Step 3: Substitute into Quadratic Formula
Substitute coefficients a, b, and c into the quadratic formula. This results in \(x = \frac{-1\pm\sqrt{1^{2}-4(1)(-2)}}{2\cdot1}\).
4Step 4: Simplify the Equation
Simplify under the square root first to get \( x = \frac{-1\pm\sqrt{1-(-8)}}{2}\). Then simplify the square root to get \( x = \frac{-1\pm\sqrt{9}}{2}\). The square root of 9 is ±3, which leads to \( x = \frac{-1\pm3}{2}\).
5Step 5: Calculate Final Values of x
The final values of x are \( x = \frac{-1+3}{2}\) which results to \(x=1\) and \( x = \frac{-1-3}{2}\) which results to \(x=-2\).
Key Concepts
Understanding Standard FormThe Quadratic Formula DecodedEffective Techniques for Solving Equations
Understanding Standard Form
When working with quadratic equations, it's essential to first express the equation in its standard form. The standard form of a quadratic equation is written as \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants where \( a eq 0 \). This format is crucial because it allows for easier identification of the coefficients which are used in solving the equation using other methods, such as the quadratic formula.
To convert an equation to standard form, rearrange all terms on one side so the equation equals zero. For instance, given the equation \( -2 + x^2 = -x \), move \( -x \) to the left side to join other terms, resulting in \( x^2 + x - 2 = 0 \). This new equation is now in standard form with coefficients \( a = 1 \), \( b = 1 \), and \( c = -2 \).
By ensuring your equation is in standard form, you set the foundation for solving it efficiently, using further techniques like the quadratic formula. This step simplifies and streamlines the problem, making it more approachable.
To convert an equation to standard form, rearrange all terms on one side so the equation equals zero. For instance, given the equation \( -2 + x^2 = -x \), move \( -x \) to the left side to join other terms, resulting in \( x^2 + x - 2 = 0 \). This new equation is now in standard form with coefficients \( a = 1 \), \( b = 1 \), and \( c = -2 \).
By ensuring your equation is in standard form, you set the foundation for solving it efficiently, using further techniques like the quadratic formula. This step simplifies and streamlines the problem, making it more approachable.
The Quadratic Formula Decoded
The quadratic formula is a powerful tool that helps us solve any quadratic equation. The formula itself is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to find the values of \( x \) that satisfy the quadratic equation \( ax^2 + bx + c = 0 \).
Once your equation is in the standard form, identify the coefficients \( a \), \( b \), and \( c \) from the equation. In our example, after converting to standard form, we have \( a = 1 \), \( b = 1 \), and \( c = -2 \). Substitute these values into the quadratic formula.
For instance, replacing our example’s coefficients results in:\[x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-2)}}{2 \cdot 1}\]
Computing the expression inside the square root yields \( 1 + 8 = 9 \), and the square root of 9 is 3. This leads to the formula simplifying to two possible solutions because of the ± sign:
Once your equation is in the standard form, identify the coefficients \( a \), \( b \), and \( c \) from the equation. In our example, after converting to standard form, we have \( a = 1 \), \( b = 1 \), and \( c = -2 \). Substitute these values into the quadratic formula.
For instance, replacing our example’s coefficients results in:\[x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-2)}}{2 \cdot 1}\]
Computing the expression inside the square root yields \( 1 + 8 = 9 \), and the square root of 9 is 3. This leads to the formula simplifying to two possible solutions because of the ± sign:
- \( x = \frac{-1+3}{2} = 1 \)
- \( x = \frac{-1-3}{2} = -2 \)
Effective Techniques for Solving Equations
Solving equations, especially quadratic ones, can seem daunting at first. However, using structured approaches simplifies this task. Start by arranging your problem into a standard form, which sets the groundwork for applying a variety of solving techniques.
In many cases, the quadratic formula stands out as a reliable method for solving quadratic equations, as it provides a clear path to finding solutions. It works for both real and complex roots, making it versatile for any type of quadratic equation.
After applying the quadratic formula and solving for the roots, it's good practice to verify your solutions by substituting back into the original equation to ensure correctness. For example, substituting \( x = 1 \) and \( x = -2 \) back into \( x^2 + x - 2 = 0 \) should satisfy the equation fully, ensuring no errors occurred during calculation.
Solving equations not only requires arithmetic skill but also a clear understanding of the steps and methods involved. As you practice, these techniques will become a natural part of your problem-solving toolkit. This way, tackling quadratic equations becomes less of a challenge and more of a straightforward process.
In many cases, the quadratic formula stands out as a reliable method for solving quadratic equations, as it provides a clear path to finding solutions. It works for both real and complex roots, making it versatile for any type of quadratic equation.
After applying the quadratic formula and solving for the roots, it's good practice to verify your solutions by substituting back into the original equation to ensure correctness. For example, substituting \( x = 1 \) and \( x = -2 \) back into \( x^2 + x - 2 = 0 \) should satisfy the equation fully, ensuring no errors occurred during calculation.
Solving equations not only requires arithmetic skill but also a clear understanding of the steps and methods involved. As you practice, these techniques will become a natural part of your problem-solving toolkit. This way, tackling quadratic equations becomes less of a challenge and more of a straightforward process.
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