Content

• Steps
• Method 1 of 3: Factoring an Equation
• Method 2 of 3: Using the quadratic formula
• Method 3 of 3: Completing the Square
• Tips

A quadratic equation is an equation in which the largest power of a variable is 2. There are three main ways to solve quadratic equations: if possible, factor the quadratic equation, use the quadratic formula, or complete the square. Do you want to know how all this is done? Read on.

Steps

Method 1 of 3: Factoring an Equation

1. 1 Add all the similar elements and transfer them to one side of the equation. This will be the first step, meaning ${ displaystyle x ^ {2}}$ in this case, it should remain positive. Add or subtract all values ${ displaystyle x ^ {2}}$, ${ displaystyle x}$ and constant, transferring everything to one part and leaving 0 in the other. Here's how to do it:
   • ${ displaystyle 2x ^ {2} -8x-4 = 3x-x ^ {2}}$
   • ${ displaystyle 2x ^ {2} + x ^ {2} -8x-3x-4 = 0}$
   • ${ displaystyle 3x ^ {2} -11x-4 = 0}$
2. 2 Factor the expression. To do this, you need to use the values ${ displaystyle x ^ {2}}$ (3), constant values ​​(-4), they must be multiplied and form -11. Here's how to do it:
   • ${ displaystyle 3x ^ {2}}$ has only two possible factors: ${ displaystyle 3x}$ and ${ displaystyle x}$so they can be written in parentheses: ${ displaystyle (3x pm?) (x pm?) = 0}$.
   • Next, substituting the factors of 4, we find the combination that, when multiplied, gives -11x. You can use a combination of 4 and 1, or 2 and 2, since both give 4. Remember that the values ​​must be negative, because we have -4.
   • Through trial and error, you get the combination ${ displaystyle (3x + 1) (x-4)}$... When multiplying, we get ${ displaystyle 3x ^ {2} -12x + x-4}$... By connecting ${ displaystyle -12x}$ and ${ displaystyle x}$, we get the middle term ${ displaystyle -11x}$which we were looking for. The quadratic equation is factorized.
   • For example, let's try an unsuitable combination: (${ displaystyle (3x-2) (x + 2)}$ = ${ displaystyle 3x ^ {2} + 6x-2x-4}$... Combining, we get ${ displaystyle 3x ^ {2} -4x-4}$... Although the factors -2 and 2 multiply to -4, the middle term does not work, because we wanted to get ${ displaystyle -11x}$, but not ${ displaystyle -4x}$.
3. 3 Equate each expression in parentheses to zero (as separate equations). This is how we find two meanings ${ displaystyle x}$for which the whole equation is equal to zero, ${ displaystyle (3x + 1) (x-4)}$ = 0. Now it remains to equate to zero each of the expressions in parentheses. Why? The point is that the product is equal to zero when at least one of the factors is equal to zero. As ${ displaystyle (3x + 1) (x-4)}$ is zero, then either (3x + 1) or (x - 4) is zero. Write down ${ displaystyle 3x + 1 = 0}$ and ${ displaystyle x-4 = 0}$.
4. 4 Solve each equation separately. In a quadratic equation, x has two meanings. Solve the equations and write down the x values:
   • Solve the equation 3x + 1 = 0
     • 3x = -1 ..... by subtracting
     • 3x / 3 = -1/3 ..... by dividing
     • x = -1/3 ..... after simplification
   • Solve the equation x - 4 = 0
     • x = 4 ..... by subtracting
   • x = (-1/3, 4) ..... possible values, i.e. x = -1/3 or x = 4.
5. 5 Check x = -1/3 by plugging this value into (3x + 1) (x - 4) = 0:
   • (3 [-1/3] + 1) ([- 1/3] - 4)? =? 0 ..... by substitution
   • (-1 + 1) (- 4 1/3)? =? 0 ..... after simplification
   • (0) (- 4 1/3) = 0 ..... after multiplication
   • 0 = 0, so x = -1/3 is the correct answer.
6. 6 Check x = 4 by plugging this value into (3x + 1) (x - 4) = 0:
   • (3 [4] + 1) ([4] - 4)? =? 0 ..... by substitution
   • (13) (4 - 4)? =? 0 ..... after simplification
   • (13) (0) = 0 ..... after multiplication
   • 0 = 0, therefore x = 4 is the correct answer.
   • Thus, both solutions are correct.

Method 2 of 3: Using the quadratic formula

1. 1 Combine all the terms and write down on one side of the equation. Save the value ${ displaystyle x ^ {2}}$ positive. Write the terms in decreasing order, thus the term ${ displaystyle x ^ {2}}$ spelled first, then ${ displaystyle x}$ and then a constant:
   • 4x - 5x - 13 = x -5
   • 4x - x - 5x - 13 +5 = 0
   • 3x - 5x - 8 = 0
2. 2 Write down the formula for the roots of a quadratic equation. The formula looks like this: ${ displaystyle { frac {-b pm { sqrt {b ^ {2} -4ac}}} {2a}}}$
3. 3 Determine the values ​​of a, b, and c in a quadratic equation. Variable a is the coefficient of the term x, b - member x, c - constant. For equation 3x -5x - 8 = 0, a = 3, b = -5, and c = -8. Write it down.
4. 4 Plug in the values ​​for a, b, and c into the equation. Knowing the values ​​of the three variables, you can plug them into the equation as follows:
   • {-b +/- √ (b - 4ac)} / 2
   • {-(-5) +/-√ ((-5) - 4(3)(-8))}/2(3) =
   • {-(-5) +/-√ ((-5) - (-96))}/2(3)
5. 5 Count it up. Substitute the values, simplify the pros and cons, and multiply or square the remaining terms:
   • {-(-5) +/-√ ((-5) - (-96))}/2(3) =
   • {5 +/-√(25 + 96)}/6
   • {5 +/-√(121)}/6
6. 6 Simplify the square root. If the square root is a square, you get an integer. If not, simplify it to the simplest root value. If the number is negative, and you are sure it must be negative, then the roots will be complex. In this example √ (121) = 11. You can write that x = (5 +/- 11) / 6.
7. 7 Find positive and negative solutions. If you've removed the square root sign, you can continue until you find positive and negative x values. Having (5 +/- 11) / 6, we can write:
   • (5 + 11)/6
   • (5 - 11)/6
8. 8 Find positive and negative values. Just count:
   • (5 + 11)/6 = 16/6
   • (5-11)/6 = -6/6
9. 9 Simplify. To do this, simply divide both by the greatest common factor. Divide the first fraction by 2, the second by 6, x is found.
   • 16/6 = 8/3
   • -6/6 = -1
   • x = (-1, 8/3)

Method 3 of 3: Completing the Square

1. 1 Move all terms to one side of the equation.a or x must be positive. This is done like this:
   • 2x - 9 = 12x =
   • 2x - 12x - 9 = 0
     • In this equation a: 2, b: -12,c: -9.
2. 2 Transfer member c (permanent) to the other side. A constant is a term in an equation that contains only a numeric value, without variables.Move it to the right side:
   • 2x - 12x - 9 = 0
   • 2x - 12x = 9
3. 3 Divide both parts by factor a or x. If x has no coefficient, then it is equal to one and this step can be skipped. In our example, we divide all members by 2:
   • 2x / 2 - 12x / 2 = 9/2 =
   • x - 6x = 9/2
4. 4 Divide b by 2, square and add to both sides. In our example b equals -6:
   • -6/2 = -3 =
   • (-3) = 9 =
   • x - 6x + 9 = 9/2 + 9
5. 5 Simplify both sides. Square the terms on the left to get (x-3) (x-3), or (x-3). Add the terms to the right to make 9/2 + 9, or 9/2 + 18/2, which is 27/2.
6. 6 Extract the square root of both sides. The square root of (x-3) is simply (x-3). The square root of 27/2 can be written as ± √ (27/2). Thus, x - 3 = ± √ (27/2).
7. 7 Simplify radical expression and find x. To simplify ± √ (27/2), find the perfect square in the numbers 27 and 2, or their factors. In 27 there is a complete square of 9, because 9 x 3 = 27. To deduce 9 from the root sign, take the root from it and subtract 3 from the root sign. Leave 3 in the numerators of the fraction under the root sign, since this factor cannot be extracted, and also leave 2 at the bottom. Next, move the constant 3 from the left side of the equation to the right side and write down the two solutions for x:
   • x = 3 + (√6) / 2
   • x = 3 - (√6) / 2)

Tips

• If the number under the root sign is not a complete square, then the last few steps are performed slightly differently. Here's an example:
• As you can see, the root sign hasn't disappeared. In this way, the terms in the numerators cannot be combined. Then there is no point in splitting the plus or minus. Instead, we divide any common factors - but only if the factor common to the constant and root coefficient.