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Magazine Chapter 9: Problem 41
Solve by (a) Completing the square (b) Using the quadratic formula $$ 7 x^{2}=x+8 $$
Short Answer
Expert verified
Answer: The solutions to the given quadratic equation are \(x = \frac{1}{14} + \sqrt{\frac{57}{49}}\) and \(x = \frac{1}{14} - \sqrt{\frac{57}{49}}\).
Step by step solution
01
Rewrite the equation in the standard form
Write the given quadratic equation in the standard form (quadratic function), which is:
$$
ax^2 + bx + c = 0
$$
Subtract x and 8 from both sides of the given equation:
$$
7x^2 - x - 8 = 0
$$
02
Completing the square
First, rewrite the equation as follows (divide by 7):
$$
x^2 - \frac{1}{7}x - \frac{8}{7} = 0
$$
Next, add and subtract \((b/2a)^2\) inside the equation, where \(a=1\) and \(b=-\frac{1}{7}\):
$$
x^2 - \frac{1}{7}x + \left(\frac{1}{14}\right)^2 - \left(\frac{1}{14}\right)^2 - \frac{8}{7} = 0
$$
Now, rewrite the equation as the square of a binomial and simplify:
$$
\left(x - \frac{1}{14}\right)^2 - \frac{57}{49} = 0
$$
03
Solve for x
Isolate x by adding \(\frac{57}{49}\) to both sides and taking the square root:
$$
\left(x - \frac{1}{14}\right)^2 = \frac{57}{49}
$$
$$
x - \frac{1}{14} = \pm\sqrt{\frac{57}{49}}
$$
Finally, add \(\frac{1}{14}\) to both sides:
$$
x = \frac{1}{14} \pm \sqrt{\frac{57}{49}}
$$
So, the solutions are:
$$
x = \frac{1}{14} + \sqrt{\frac{57}{49}}
$$
and
$$
x = \frac{1}{14} - \sqrt{\frac{57}{49}}
$$
#b) Using the quadratic formula#
04
Identify a, b, and c from the given equation
The given quadratic equation is:
$$
7x^2 - x - 8 = 0
$$
Compare it with the standard form, which is:
$$
ax^2 + bx + c = 0
$$
We can identify that:
$$
a = 7, b = -1, c = -8
$$
05
Apply the quadratic formula
The quadratic formula is given as:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
Plug in the values of a, b, and c:
$$
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(7)(-8)}}{2(7)}
$$
06
Simplify and solve for x
Simplify and solve for x:
$$
x = \frac{1 \pm \sqrt{1 + 224}}{14}
$$
$$
x = \frac{1 \pm \sqrt{225}}{14}
$$
$$
x = \frac{1 \pm 15}{14}
$$
So, the solutions are:
$$
x = \frac{1 + 15}{14}
$$
and
$$
x = \frac{1 - 15}{14}
$$
Comparing the results from both methods, we can see that the solutions found are equivalent.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method for solving quadratic equations by transforming a standard form equation into a perfect square trinomial. This procedure involves a few systematic steps that make it easy to find the roots of the equation.
- First, ensure the quadratic is in the form of \(ax^2 + bx + c = 0\). If \(a eq 1\), divide the entire equation by \(a\) to make the leading coefficient 1, easing further calculations.
- Identify the coefficient of the linear term (\(b\)) and divide it by 2, then square the result. This value is added and subtracted within the equation to form a complete square.
- The equation now can be rewritten using the identity: \((x + p)^2 = q\), making it easy to solve for \(x\) by taking the square root of both sides.
Quadratic Formula
The quadratic formula provides a universal solution to any quadratic equation that can be expressed as \(ax^2 + bx + c = 0\). This powerful formula is particularly useful when other methods, such as factoring, are not viable.
The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
- Identify: From the quadratic equation, identify the coefficients \(a\), \(b\), and \(c\).
- Substitute: Insert these values into the quadratic formula.
- Calculate: Simplify under the square root, known as the discriminant \(b^2 - 4ac\). This step indicates the nature of the roots (real and distinct, real and repeated, or complex).
- Solve: Calculate the two potential solutions from the equation.
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is an essential framework that fosters clarity and consistency in solving these equations. Expressed as \(ax^2 + bx + c = 0\), it allows mathematicians to develop structured procedures for finding solutions.
- Structure: Ensuring the equation is in standard form is foundational for applying methods like completing the square or using the quadratic formula effectively. Both approaches heavily rely on this format.
- Initial Setup: To set your equation in standard form, rearrange any given quadratic equation by moving all terms to one side, ensuring the equation equals zero.
- Clarity: Identifying \(a\), \(b\), and \(c\) from the equation is crucial, as these components are directly used in various methods to find solutions.
