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The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \). The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps us compute the roots (solutions) of any quadratic equation, even those that cannot be factored easily.

• Identify coefficients: Find \(a\), \(b\), and \(c\) in your equation.
• Calculate the discriminant: \(b^2 - 4ac\). This part of the formula determines how many solutions exist and their nature (real or complex).
• Use the formula: Plug \(a\), \(b\), and \(c\) into the formula for \(x\). Simplify to find the possible solutions.

Even if you factor an equation, checking with the quadratic formula can confirm your solutions. This comes in handy especially when the solutions are not integers or the equation cannot be easily factored.

Factoring quadratic equations is a method to express a quadratic equation as a product of its roots. For example, a quadratic expressed as \( (x + m)(x + n) = 0 \) implies that the roots are \(-m\) and \(-n\). This method is often quicker than using the quadratic formula, but requires the equation to be factorable.When trying to factor a quadratic like \( t^2 + 3t - 10 = 0 \), look for two numbers that multiply to give you the constant term, \( -10 \), and add to give you the linear coefficient, \( 3 \). In this case, the numbers \( 5 \) and \( -2 \) work. Hence, you can express the quadratic as \[(t + 5)(t - 2) = 0\]

• Understand the format: Aim to express the quadratic in the form \((t + p)(t + q) = 0\).
• Find the factors: Search for number pairs like \(5\) and \(-2\) that satisfy the conditions.
• Write it as a product: Once you find appropriate values, rewrite the equation.

This factorization helps find solutions quickly by setting each bracket equal to zero. Solving \((t + 5) = 0\) and \((t - 2) = 0\) gives the roots; here, these are \(t = -5\) and \(t = 2\).

After finding solutions to a quadratic equation, it's important to check them for accuracy. This validation step ensures the solutions satisfy the original equation.Here’s how you check each solution:

• Plug the solution back into the original equation wherever the variable appears.
• Simplify both sides to see if they equate.
• If both sides of the equation are equal, the solution is correct.

For example, with our equation \(\frac{10}{t}-t=3\), check \(t = -5\):\[\frac{10}{-5} - (-5) = -2 + 5 = 3\]This matches the right side of the equation, confirming \(t = -5\) as a valid solution.Similarly, for \(t = 2\):\[\frac{10}{2} - 2 = 5 - 2 = 3\]Here again, both sides are equal, confirming \(t = 2\) is correct.Always check each potential solution to avoid mistakes in handling the initial equation or during factoring and simplifications!