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Magazine Chapter 1: Problem 71
Solve by using the quadratic formula. \(0.4 y^{2}=2 y-2.5\)
Short Answer
Expert verified
The solution is y = 2.5.
Step by step solution
01
Arrange the equation in standard form
Rewrite the given equation in the standard form of a quadratic equation, which is y^2 + by + c = 0Let's move all terms to one side of the equation:0.4y^2 - 2y + 2.5 = 0
02
Identify coefficients
Identify the coefficients 'a', 'b', and 'c' from the standard form equation. In this case:a = 0.4b = -2c = 2.5
03
Write the quadratic formula
The quadratic formula \[y = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]will be used to find the roots of the quadratic equation.
04
Plug in the coefficients
Substitute the identified coefficients into the quadratic formula:\[y = \frac{{-(-2) \pm \sqrt{{(-2)^2 - 4(0.4)(2.5)}}}}{{2(0.4)}}\]
05
Simplify under the square root
Simplify the expression under the square root (the discriminant):\[y = \frac{{2 \pm \sqrt{{4 - 4(1)}}}}{{0.8}}\]\[y = \frac{{2 \pm \sqrt{{4 - 4}}}}{{0.8}}\]\[y = \frac{{2 \pm \sqrt{0}}}{{0.8}}\]\[y = \frac{{2 \pm 0}}{{0.8}}\]
06
Solve for y
Since the square root part is zero, the quadratic formula simplifies to one solution:\[y = \frac{2}{0.8}\]\[y = 2.5\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
quadratic formula
The quadratic formula is a powerful tool for solving any quadratic equation. It can be used when a quadratic equation is in the standard form and other methods, like factoring, do not work easily.
The quadratic formula is given by: \[y = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]
This formula gives the values of \( y \) that are the solutions to the quadratic equation.
Key points to remember about the quadratic formula:
The quadratic formula is given by: \[y = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]
This formula gives the values of \( y \) that are the solutions to the quadratic equation.
Key points to remember about the quadratic formula:
- The part under the square root, called the discriminant, is critical. It determines the nature and number of solutions.
- The discriminant is the expression \( b^2 - 4ac \).
- The entire quadratic formula is derived from the standard form of a quadratic equation \( ay^2 + by + c = 0 \).
standard form of quadratic equation
Quadratic equations need to be in their standard form to use the quadratic formula effectively.
The standard form of a quadratic equation is: \( ay^2 + by + c = 0 \)
This arrangement allows us to clearly identify the coefficients and use them in the quadratic formula.
The standard form of a quadratic equation is: \( ay^2 + by + c = 0 \)
- In this form, 'a,' 'b,' and 'c' are constants, and 'a' must not be zero (since if 'a' were zero, the equation wouldn't be quadratic anymore).
- Putting the equation into this form involves rearranging the terms so that one side of the equation equals zero. For example,\(0.4 y^{2}=2 y-2.5\) becomes \(0.4y^2 - 2y + 2.5 = 0 \).
This arrangement allows us to clearly identify the coefficients and use them in the quadratic formula.
discriminant calculation
The discriminant is a key component of the quadratic formula and gives us valuable information about the solutions. The discriminant is calculated using: \[ \text{Discriminant} = b^2 - 4ac \]
Here鈥檚 what the discriminant tells us:
In our example, the discriminant calculation is: \[-2^2 - 4(0.4)(2.5)\]
Simplifying this we get: \[4 - 4(0.4)(2.5)\]
Which simplifies to: \[4 - 4 = 0\]
Therefore, there is exactly one real solution.
Here鈥檚 what the discriminant tells us:
- If the discriminant is positive (>0), the quadratic equation has two distinct real solutions.
- If the discriminant is zero (=0), the equation has exactly one real solution.
- If the discriminant is negative (<0), the quadratic equation has no real solutions, but two complex solutions.
In our example, the discriminant calculation is: \[-2^2 - 4(0.4)(2.5)\]
Simplifying this we get: \[4 - 4(0.4)(2.5)\]
Which simplifies to: \[4 - 4 = 0\]
Therefore, there is exactly one real solution.
coefficients identification
Identifying coefficients is crucial for applying the quadratic formula correctly. The coefficients are constants from the terms of the quadratic equation \(ay^2 + by +c = 0\).
Here鈥檚 how to identify them:
For the given equation 0.4y^2 - 2y + 2.5 = 0, we identify:
Here鈥檚 how to identify them:
- 'a' is the coefficient of \(y^2\) term.
- 'b' is the coefficient of the \(y\) term.
- 'c' is the constant term.
For the given equation 0.4y^2 - 2y + 2.5 = 0, we identify:
- \(a = 0.4\)
- \(b = -2\)
- \(c = 2.5\)
