Question

Find the real solutions, if any, of each equation. $$ 3 x^{2}+7 x-20=0 $$

Short answer

The real solutions are \( x = \frac{5}{3} \) and \( x = -4 \).

Step-by-step solution

Identify the coefficients

In the quadratic equation, identify the coefficients, where the equation is in the form of \(ax^2 + bx + c = 0\). Here, \(a = 3\), \(b = 7\), and \(c = -20\).

Apply the quadratic formula

Use the quadratic formula to find the solutions of the equation: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 3\), \(b = 7\), and \(c = -20\) into the formula.

Calculate the discriminant

Compute the discriminant, \(D = b^2 - 4ac\). Here, \( D = 7^2 - 4 \cdot 3 \cdot (-20)\), thus, \(D = 49 + 240 = 289\).

Calculate the solutions

Since the discriminant is positive, there are real solutions. Use the quadratic formula to find these: \(x = \frac{-7 \pm \sqrt{289}}{6}\), hence, \(x = \frac{-7 \pm 17}{6}\). This gives two solutions: \(x_1 = \frac{-7 + 17}{6} = \frac{10}{6} = \frac{5}{3}\), and \(x_2 = \frac{-7 - 17}{6} = \frac{-24}{6} = -4\).

Key concepts

quadratic formula

When you're solving quadratic equations, one of the most reliable tools you can use is the quadratic formula. The formula is written as:

• First, identify the coefficients from your quadratic equation. These coefficients are typically denoted as 'a', 'b', and 'c'.
  In our example, we have the equation:
  
  
  
  
  
  
  
• Once you've identified 'a', 'b', and 'c', you can plug them into the quadratic formula:
  
  
  
  
  
• Finally, simplify the results to find the solutions to the quadratic equation.
  For our example, after plugging in the coefficients, we would continue to simplify until we find the values of 'x' that satisfy the equation.

coefficients

Understanding coefficients is crucial when working with quadratic equations. Coefficients are the numerical factors in front of the variables in an equation. In a standard form quadratic equation, which looks like this:

• 'a' is the coefficient of the term with squared variable, which in our example is 3.
  
  
• 'b' is the coefficient of the term with the single variable, which is 7.
  
  
• 'c' is the constant term, in this case, -20.

Each of these coefficients plays a role in the quadratic formula. Accurately identifying and substituting them into the formula can greatly simplify the problem-solving process. For our given equation:





Remember:

• Consistency in identifying 'a', 'b', and 'c'.
  
  
  
• Correctly substituting them into the quadratic formula.
  
  
• Carefully performing the arithmetic operations.
  

discriminant

The discriminant, represented by 'D', is a key part of the quadratic formula. The discriminant helps determine the nature of the roots of the quadratic equation. The formula for the discriminant is:

• In our example, where 'b' is 7, 'a' is 3, and 'c' is -20, the discriminant is calculated as follows:
  
  
  
  
• The value of the discriminant is 289.

The value of the discriminant tells us:

• If 'D' is positive, the quadratic equation has two distinct real roots.
  
• If 'D' is zero, there is exactly one real root.
  
• If 'D' is negative, there are no real roots; instead, there are two complex roots.

In our solved example, the discriminant is 289, a positive value, indicating two distinct real roots. Sure enough, when we solved the equation, we got:

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