first, lets solve by factoring:
Let's solve your equation step-by-step.
2x2+14x−4=−x2+3x
Step 1: Subtract -x^2+3x from both sides.
2x2+14x−4−(−x2+3x)=−x2+3x−(−x2+3x)
3x2+11x−4=0
Step 2: Factor left side of equation.
(3x−1)(x+4)=0
Step 3: Set factors equal to 0.
3x−1=0 or x+4=0
x=
1
3
or x=−4
we can also solve using the quadratic formula:
2x2+14x−4=−x2+3x
Step 1: Subtract -x^2+3x from both sides.
2x2+14x−4−(−x2+3x)=−x2+3x−(−x2+3x)
3x2+11x−4=0
Step 2: Use quadratic formula with a=3, b=11, c=-4.
x=
−b±√b2−4ac
2a
x=
−(11)±√(11)2−4(3)(−4)
2(3)
x=
−11±√169
6
x=
1
3
or x=−4
lastly, we can complete the square.
2x2+14x−4=−x2+3x
Step 1: Add x^2 to both sides.
2x2+14x−4+x2=−x2+3x+x2
3x2+14x−4=3x
Step 2: Subtract 3x from both sides.
3x2+14x−4−3x=3x−3x
3x2+11x−4=0
Step 3: Add 4 to both sides.
3x2+11x−4+4=0+4
3x2+11x=4
Step 4: Since the coefficient of 3x^2 is 3, divide both sides by 3.
3x2+11x
3
=
4
3
x2+
11
3
x=
4
3
Step 5: The coefficient of 11/3x is 11/3. Let b=11/3.
Then we need to add (b/2)^2=121/36 to both sides to complete the square.
Add 121/36 to both sides.
x2+
11
3
x+
121
36
=
4
3
+
121
36
x2+
11
3
x+
121
36
=
169
36
Step 6: Factor left side.
(x+
11
6
)2=
169
36
Step 7: Take square root.
x+
11
6
=±√
169
36
Step 8: Add (-11)/6 to both sides.
x+
11
6
+
−11
6
=
−11
6
±√
169
36
x=
−11
6
±√
169
36
x=
1
3
or x=−4
Answer:
x=
1
3
or x=−4
